Theorem cbvabw 2262

Description: Version of cbvab 2263 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem cbvabw

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvabw.1	. . . . 5 |- F/ y ph
2	1	sbco2v 1921	. . . 4 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
3		cbvabw.2	. . . . . 6 |- F/ x ps
4		cbvabw.3	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
5	3, 4	sbiev 1765	. . . . 5 |- ( [ y / x ] ph <-> ps )
6	5	sbbii 1738	. . . 4 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] ps )
7	2, 6	bitr3i 185	. . 3 |- ( [ z / x ] ph <-> [ z / y ] ps )
8		df-clab 2126	. . 3 |- ( z e. { x | ph } <-> [ z / x ] ph )
9		df-clab 2126	. . 3 |- ( z e. { y | ps } <-> [ z / y ] ps )
10	7, 8, 9	3bitr4i 211	. 2 |- ( z e. { x | ph } <-> z e. { y | ps } )
11	10	eqriv 2136	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   F/wnf 1436   e. wcel 1480   [wsb 1735   {cab 2125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132

This theorem is referenced by:  cbvsbcw  2936