Theorem cbvab 2746

Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)

Hypotheses

Ref	Expression
cbvab.1	|- F/ y ph
cbvab.2	|- F/ x ps
cbvab.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Proof of Theorem cbvab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvab.1	. . . . 5 |- F/ y ph
2	1	sbco2 2415	. . . 4 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
3		cbvab.2	. . . . . 6 |- F/ x ps
4		cbvab.3	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
5	3, 4	sbie 2408	. . . . 5 |- ( [ y / x ] ph <-> ps )
6	5	sbbii 1887	. . . 4 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] ps )
7	2, 6	bitr3i 266	. . 3 |- ( [ z / x ] ph <-> [ z / y ] ps )
8		df-clab 2609	. . 3 |- ( z e. { x | ph } <-> [ z / x ] ph )
9		df-clab 2609	. . 3 |- ( z e. { y | ps } <-> [ z / y ] ps )
10	7, 8, 9	3bitr4i 292	. 2 |- ( z e. { x | ph } <-> z e. { y | ps } )
11	10	eqriv 2619	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   F/wnf 1708   [wsb 1880   e. wcel 1990   {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615

This theorem is referenced by:  cbvabv  2747  cbvrab  3198  cbvsbc  3464  cbvrabcsf  3568  rabsnifsb  4257  dfdmf  5317  dfrnf  5364  funfv2f  6267  abrexex2g  7144  abrexex2OLD  7150  bnj873  30994  ptrest  33408  poimirlem26  33435  poimirlem27  33436