Theorem nn1suc 4435

Description: If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor.

Hypotheses

Ref	Expression
nn1suc.1	|- (x = 1 -> (ph <-> ps))
nn1suc.3	|- (x = (y + 1) -> (ph <-> ch))
nn1suc.4	|- (x = A -> (ph <-> th))
nn1suc.5	|- ps
nn1suc.6	|- (y e. NN -> ch)

Assertion

Ref	Expression
nn1suc	|- (A e. NN -> th)

Distinct variable group(s):   x,y,A   ps,x   ch,x   th,x   ph,y

Proof of Theorem nn1suc

Step	Hyp	Ref	Expression
1		dfsbcq 1442	. . 3 |- (z = 1 -> ([z / x](A e. NN -> ph) <-> [1 / x](A e. NN -> ph)))
2		sbequ 877	. . 3 |- (z = y -> ([z / x](A e. NN -> ph) <-> [y / x](A e. NN -> ph)))
3		dfsbcq 1442	. . 3 |- (z = (y + 1) -> ([z / x](A e. NN -> ph) <-> [(y + 1) / x](A e. NN -> ph)))
4		dfsbcq 1442	. . . . . . . 8 |- (z = A -> ([z / x]ph <-> [A / x]ph))
5		elex 1356	. . . . . . . . . . 11 |- (A e. NN -> E.x x = A)
6		ax-17 925	. . . . . . . . . . . . . 14 |- (z e. A -> A.x z e. A)
7	6	hbsbc 1446	. . . . . . . . . . . . 13 |- ((A e. NN -> [A / x]ph) -> A.x(A e. NN -> [A / x]ph))
8		ax-17 925	. . . . . . . . . . . . 13 |- ((A e. NN -> th) -> A.x(A e. NN -> th))
9	7, 8	hbbi 705	. . . . . . . . . . . 12 |- (((A e. NN -> [A / x]ph) <-> (A e. NN -> th)) -> A.x((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
10		sbceq1 1443	. . . . . . . . . . . . . 14 |- (x = A -> (ph <-> [A / x]ph))
11		nn1suc.4	. . . . . . . . . . . . . 14 |- (x = A -> (ph <-> th))
12	10, 11	bitr3d 408	. . . . . . . . . . . . 13 |- (x = A -> ([A / x]ph <-> th))
13	12	imbi2d 464	. . . . . . . . . . . 12 |- (x = A -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
14	9, 13	19.23ai 746	. . . . . . . . . . 11 |- (E.x x = A -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
15	5, 14	syl 12	. . . . . . . . . 10 |- (A e. NN -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
16	15	pm5.74rd 446	. . . . . . . . 9 |- (A e. NN -> (A e. NN -> ([A / x]ph <-> th)))
17	16	pm2.43i 58	. . . . . . . 8 |- (A e. NN -> ([A / x]ph <-> th))
18	4, 17	sylan9bbr 419	. . . . . . 7 |- ((A e. NN /\ z = A) -> ([z / x]ph <-> th))
19	18	exp 291	. . . . . 6 |- (A e. NN -> (z = A -> ([z / x]ph <-> th)))
20	19	com12 13	. . . . 5 |- (z = A -> (A e. NN -> ([z / x]ph <-> th)))
21	20	pm5.74d 444	. . . 4 |- (z = A -> ((A e. NN -> [z / x]ph) <-> (A e. NN -> th)))
22		ax-17 925	. . . . 5 |- (A e. NN -> A.x A e. NN)
23	22	sb19.21 888	. . . 4 |- ([z / x](A e. NN -> ph) <-> (A e. NN -> [z / x]ph))
24	21, 23	syl5bb 410	. . 3 |- (z = A -> ([z / x](A e. NN -> ph) <-> (A e. NN -> th)))
25		1nn 4432	. . . . . . . 8 |- 1 e. NN
26	25	elisseti 1355	. . . . . . 7 |- 1 e. V
27	26	isseti 1352	. . . . . 6 |- E.x x = 1
28	26	hbsbcv 1447	. . . . . . 7 |- ([1 / x]ph -> A.x[1 / x]ph)
29		nn1suc.5	. . . . . . . . 9 |- ps
30		nn1suc.1	. . . . . . . . 9 |- (x = 1 -> (ph <-> ps))
31	29, 30	mpbiri 169	. . . . . . . 8 |- (x = 1 -> ph)
32		sbceq1 1443	. . . . . . . 8 |- (x = 1 -> (ph <-> [1 / x]ph))
33	31, 32	mpbid 170	. . . . . . 7 |- (x = 1 -> [1 / x]ph)
34	28, 33	19.23ai 746	. . . . . 6 |- (E.x x = 1 -> [1 / x]ph)
35	27, 34	ax-mp 6	. . . . 5 |- [1 / x]ph
36	35	a1i 7	. . . 4 |- (A e. NN -> [1 / x]ph)
37	22	sbc19.21g 1470	. . . . 5 |- (1 e. V -> ([1 / x](A e. NN -> ph) <-> (A e. NN -> [1 / x]ph)))
38	26, 37	ax-mp 6	. . . 4 |- ([1 / x](A e. NN -> ph) <-> (A e. NN -> [1 / x]ph))
39	36, 38	mpbir 165	. . 3 |- [1 / x](A e. NN -> ph)
40		nn1suc.6	. . . . . . 7 |- (y e. NN -> ch)
41		oprex 3018	. . . . . . . . 9 |- (y + 1) e. V
42	41	isseti 1352	. . . . . . . 8 |- E.x x = (y + 1)
43		ax-17 925	. . . . . . . . . 10 |- (ch -> A.xch)
44	41	hbsbcv 1447	. . . . . . . . . 10 |- ([(y + 1) / x]ph -> A.x[(y + 1) / x]ph)
45	43, 44	hbbi 705	. . . . . . . . 9 |- ((ch <-> [(y + 1) / x]ph) -> A.x(ch <-> [(y + 1) / x]ph))
46		nn1suc.3	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> ch))
47		sbceq1 1443	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> [(y + 1) / x]ph))
48	46, 47	bitr3d 408	. . . . . . . . 9 |- (x = (y + 1) -> (ch <-> [(y + 1) / x]ph))
49	45, 48	19.23ai 746	. . . . . . . 8 |- (E.x x = (y + 1) -> (ch <-> [(y + 1) / x]ph))
50	42, 49	ax-mp 6	. . . . . . 7 |- (ch <-> [(y + 1) / x]ph)
51	40, 50	sylib 173	. . . . . 6 |- (y e. NN -> [(y + 1) / x]ph)
52	51	a1d 14	. . . . 5 |- (y e. NN -> (A e. NN -> [(y + 1) / x]ph))
53	22	sbc19.21g 1470	. . . . . 6 |- ((y + 1) e. V -> ([(y + 1) / x](A e. NN -> ph) <-> (A e. NN -> [(y + 1) / x]ph)))
54	41, 53	ax-mp 6	. . . . 5 |- ([(y + 1) / x](A e. NN -> ph) <-> (A e. NN -> [(y + 1) / x]ph))
55	52, 54	sylibr 175	. . . 4 |- (y e. NN -> [(y + 1) / x](A e. NN -> ph))
56	55	a1d 14	. . 3 |- (y e. NN -> ([y / x](A e. NN -> ph) -> [(y + 1) / x](A e. NN -> ph)))
57	1, 2, 3, 24, 39, 56	nnind 4434	. 2 |- (A e. NN -> (A e. NN -> th))
58	57	pm2.43i 58	1 |- (A e. NN -> th)

Syntax hints:   -> wi 2   <-> wb 127  E.wex 678  [wsb 852   = wceq 1091   e. wcel 1092  Vcvv 1348  [wsbc 1440  (class class class)co 3001  1c1 4029   + caddc 4031  NNcn 4093

This theorem is referenced by:  ruclem29 4913

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-ltp 3884  df-plpr 3958  df-enr 3960  df-nr 3961  df-plr 3962  df-0r 3965  df-1r 3966  df-c 4034  df-1 4036  df-r 4038  df-plus 4039  df-n 4423

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