Theorem cbvab 1908

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbvab.1	|- (ph -> A.yph)
cbvab.2	|- (ps -> A.xps)
cbvab.3	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvab	|- {x | ph} = {y | ps}

Distinct variable group:   x,y

Proof of Theorem cbvab

Step	Hyp	Ref	Expression
1		cbvab.1	. . . 4 |- (ph -> A.yph)
2	1	hbab 1467	. . 3 |- (z e. {x | ph} -> A.y z e. {x | ph})
3		hbab1 1466	. . 3 |- (z e. {y | ps} -> A.y z e. {y | ps})
4	2, 3	cleqf 1560	. 2 |- ({x | ph} = {y | ps} <-> A.y(y e. {x | ph} <-> y e. {y | ps}))
5		cbvab.2	. . . 4 |- (ps -> A.xps)
6		visset 1813	. . . 4 |- y e. V
7		cbvab.3	. . . 4 |- (x = y -> (ph <-> ps))
8	5, 6, 7	elabf 1896	. . 3 |- (y e. {x | ph} <-> ps)
9		abid 1465	. . 3 |- (y e. {y | ps} <-> ps)
10	8, 9	bitr4 176	. 2 |- (y e. {x | ph} <-> y e. {y | ps})
11	4, 10	mpgbir 988	1 |- {x | ph} = {y | ps}

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  {cab 1463

This theorem is referenced by:  cbvabv 1909  cbvrab 1910  csbabg 2043  dfdmf 3306  dfrnf 3348  funfv2f 3772  abrexexlem2 3859  abrexex2 3871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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