Theorem cbvsbc 3464

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 2746	. . 3 |- { x | ph } = { y | ps }
5	4	eleq2i 2693	. 2 |- ( A e. { x | ph } <-> A e. { y | ps } )
6		df-sbc 3436	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
7		df-sbc 3436	. 2 |- ( [. A / y ]. ps <-> A e. { y | ps } )
8	5, 6, 7	3bitr4i 292	1 |- ( [. A / x ]. ph <-> [. A / y ]. ps )

Syntax hints:   -> wi 4   <-> wb 196   F/wnf 1708   e. wcel 1990   {cab 2608   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by:  cbvsbcv  3465  cbvcsb  3538