Theorem cbvsbc 2791

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 2160	. . 3 |- { x | ph } = { y | ps }
5	4	eleq2i 2104	. 2 |- ( A e. { x | ph } <-> A e. { y | ps } )
6		df-sbc 2765	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
7		df-sbc 2765	. 2 |- ( [. A / y ]. ps <-> A e. { y | ps } )
8	5, 6, 7	3bitr4i 201	1 |- ( [. A / x ]. ph <-> [. A / y ]. ps )

Syntax hints:   -> wi 4   <-> wb 98   F/wnf 1349   e. wcel 1393   {cab 2026   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765

This theorem is referenced by:  cbvsbcv  2792  cbvcsb  2856