Theorem cbvsbcw 3013

Description: Version of cbvsbc 3014 with a disjoint variable condition. (Contributed by GG, 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 2316	. . 3 |- { x | ph } = { y | ps }
5	4	eleq2i 2260	. 2 |- ( A e. { x | ph } <-> A e. { y | ps } )
6		df-sbc 2986	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
7		df-sbc 2986	. 2 |- ( [. A / y ]. ps <-> A e. { y | ps } )
8	5, 6, 7	3bitr4i 212	1 |- ( [. A / x ]. ph <-> [. A / y ]. ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1471   e. wcel 2164   {cab 2179   [.wsbc 2985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-sbc 2986

This theorem is referenced by:  cbvcsbw  3084