Theorem cbvsbc 2865

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 2210	. . 3 |- { x | ph } = { y | ps }
5	4	eleq2i 2154	. 2 |- ( A e. { x | ph } <-> A e. { y | ps } )
6		df-sbc 2839	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
7		df-sbc 2839	. 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 1394   e. wcel 1438   {cab 2074   [.wsbc 2838

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2839

This theorem is referenced by:  cbvsbcv  2866  cbvcsb  2935