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 → (φ ↔ ψ))
nn1suc.3	⊢ (x = (y + 1) → (φ ↔ χ))
nn1suc.4	⊢ (x = A → (φ ↔ θ))
nn1suc.5	⊢ ψ
nn1suc.6	⊢ (y ∈ ℕ → χ)

Assertion

Ref	Expression
nn1suc	⊢ (A ∈ ℕ → θ)

Distinct variable group(s):   x,y,A   ψ,x   χ,x   θ,x   φ,y

Proof of Theorem nn1suc

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

Syntax hints:   → wi 2   ↔ wb 127  ∃wex 678  [wsb 852   = wceq 1091   ∈ wcel 1092  Vcvv 1348  [wsbc 1440  (class class class)co 3001  1c1 4029   + caddc 4031  ℕcn 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

metamath.org