Theorem cbvsbcw 2978

Description: Version of cbvsbc 2979 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
cbvsbcw	|- ( [. A / x ]. ph <-> [. A / y ]. ps )

Distinct variable group:   x, y

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

Proof of Theorem cbvsbcw

Step	Hyp	Ref	Expression
1		cbvsbcw.1	. . . 4 |- F/ y ph
2		cbvsbcw.2	. . . 4 |- F/ x ps
3		cbvsbcw.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvabw 2289	. . 3 |- { x | ph } = { y | ps }
5	4	eleq2i 2233	. 2 |- ( A e. { x | ph } <-> A e. { y | ps } )
6		df-sbc 2952	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
7		df-sbc 2952	. 2 |- ( [. A / y ]. ps <-> A e. { y | ps } )
8	5, 6, 7	3bitr4i 211	1 |- ( [. A / x ]. ph <-> [. A / y ]. ps )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1448   e. wcel 2136   {cab 2151   [.wsbc 2951

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-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-sbc 2952

This theorem is referenced by:  cbvcsbw  3049