Theorem hbsb4 905

Description: A variable not free remains so after substitution with a distinct variable.

Hypothesis

Ref	Expression
hbsb4.1	|- (ph -> A.zph)

Assertion

Ref	Expression
hbsb4	|- (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph))

Proof of Theorem hbsb4

Step	Hyp	Ref	Expression
1		ax-8 798	. . . . . . 7 |- (x = z -> (x = y -> z = y))
2	1	a4s 682	. . . . . 6 |- (A.x x = z -> (x = y -> z = y))
3	2	eq4s 822	. . . . 5 |- (A.z z = x -> (x = y -> z = y))
4	3	del35 836	. . . 4 |- (A.z z = x -> (A.x x = y -> A.z z = y))
5	4	con3d 87	. . 3 |- (A.z z = x -> (-. A.z z = y -> -. A.x x = y))
6		hbsb2 873	. . . 4 |- (-. A.x x = y -> ([y / x]ph -> A.x[y / x]ph))
7		ax-10 800	. . . . 5 |- (A.x x = z -> (A.x[y / x]ph -> A.z[y / x]ph))
8	7	eq4s 822	. . . 4 |- (A.z z = x -> (A.x[y / x]ph -> A.z[y / x]ph))
9	6, 8	syl9r 56	. . 3 |- (A.z z = x -> (-. A.x x = y -> ([y / x]ph -> A.z[y / x]ph)))
10	5, 9	syld 27	. 2 |- (A.z z = x -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))
11		eq5 824	. . . . . 6 |- (A.x x = y -> A.zA.x x = y)
12		ax-4 673	. . . . . . 7 |- (A.x x = y -> x = y)
13	12	19.20i 691	. . . . . 6 |- (A.zA.x x = y -> A.z x = y)
14		sbequ2 864	. . . . . . . 8 |- (x = y -> ([y / x]ph -> ph))
15	14	a4s 682	. . . . . . 7 |- (A.z x = y -> ([y / x]ph -> ph))
16		sbequ1 863	. . . . . . . . 9 |- (x = y -> (ph -> [y / x]ph))
17	16	19.20ii 692	. . . . . . . 8 |- (A.z x = y -> (A.zph -> A.z[y / x]ph))
18		hbsb4.1	. . . . . . . 8 |- (ph -> A.zph)
19	17, 18	syl5 22	. . . . . . 7 |- (A.z x = y -> (ph -> A.z[y / x]ph))
20	15, 19	syld 27	. . . . . 6 |- (A.z x = y -> ([y / x]ph -> A.z[y / x]ph))
21	11, 13, 20	3syl 21	. . . . 5 |- (A.x x = y -> ([y / x]ph -> A.z[y / x]ph))
22	21	a1d 14	. . . 4 |- (A.x x = y -> ((-. A.z z = x /\ -. A.z z = y) -> ([y / x]ph -> A.z[y / x]ph)))
23		sb4 861	. . . . 5 |- (-. A.x x = y -> ([y / x]ph -> A.x(x = y -> ph)))
24		eq6 826	. . . . . . . 8 |- (-. A.z z = x -> A.x -. A.z z = x)
25		eq6 826	. . . . . . . 8 |- (-. A.z z = y -> A.x -. A.z z = y)
26	24, 25	hban 704	. . . . . . 7 |- ((-. A.z z = x /\ -. A.z z = y) -> A.x(-. A.z z = x /\ -. A.z z = y))
27		eq6 826	. . . . . . . . 9 |- (-. A.z z = x -> A.z -. A.z z = x)
28		eq6 826	. . . . . . . . 9 |- (-. A.z z = y -> A.z -. A.z z = y)
29	27, 28	hban 704	. . . . . . . 8 |- ((-. A.z z = x /\ -. A.z z = y) -> A.z(-. A.z z = x /\ -. A.z z = y))
30		ax-12 802	. . . . . . . . 9 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))
31	30	imp 277	. . . . . . . 8 |- ((-. A.z z = x /\ -. A.z z = y) -> (x = y -> A.z x = y))
32	18	a1i 7	. . . . . . . 8 |- ((-. A.z z = x /\ -. A.z z = y) -> (ph -> A.zph))
33	29, 31, 32	hbimd 787	. . . . . . 7 |- ((-. A.z z = x /\ -. A.z z = y) -> ((x = y -> ph) -> A.z(x = y -> ph)))
34	26, 33	19.20d 693	. . . . . 6 |- ((-. A.z z = x /\ -. A.z z = y) -> (A.x(x = y -> ph) -> A.xA.z(x = y -> ph)))
35		sb2 859	. . . . . . . 8 |- (A.x(x = y -> ph) -> [y / x]ph)
36	35	19.20i 691	. . . . . . 7 |- (A.zA.x(x = y -> ph) -> A.z[y / x]ph)
37	36	a7s 689	. . . . . 6 |- (A.xA.z(x = y -> ph) -> A.z[y / x]ph)
38	34, 37	syl6 23	. . . . 5 |- ((-. A.z z = x /\ -. A.z z = y) -> (A.x(x = y -> ph) -> A.z[y / x]ph))
39	23, 38	syl9 55	. . . 4 |- (-. A.x x = y -> ((-. A.z z = x /\ -. A.z z = y) -> ([y / x]ph -> A.z[y / x]ph)))
40	22, 39	pm2.61i 110	. . 3 |- ((-. A.z z = x /\ -. A.z z = y) -> ([y / x]ph -> A.z[y / x]ph))
41	40	exp 291	. 2 |- (-. A.z z = x -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))
42	10, 41	pm2.61i 110	1 |- (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672   = weq 797  [wsb 852

This theorem is referenced by:  hbsb4t 906  ddelimf 908  sbco2 913  hbsb 987  sbal1 996  hbab 1096

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

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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