Theorem cbvsbc 2842

Description: Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

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

Assertion

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

Proof of Theorem cbvsbc

Step	Hyp	Ref	Expression
1		cbvsbc.1	. . . 4 |- F/ y ph
2		cbvsbc.2	. . . 4 |- F/ x ps
3		cbvsbc.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvab 2201	. . 3 |- { x | ph } = { y | ps }
5	4	eleq2i 2145	. 2 |- ( A e. { x | ph } <-> A e. { y | ps } )
6		df-sbc 2816	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
7		df-sbc 2816	. 2 |- ( [. A / y ]. ps <-> A e. { y | ps } )
8	5, 6, 7	3bitr4i 210	1 |- ( [. A / x ]. ph <-> [. A / y ]. ps )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1389   e. wcel 1433   {cab 2067   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-sbc 2816

This theorem is referenced by:  cbvsbcv  2843  cbvcsb  2912