Theorem cbvsbc 3018

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 2320	. . 3 |- { x | ph } = { y | ps }
5	4	eleq2i 2263	. 2 |- ( A e. { x | ph } <-> A e. { y | ps } )
6		df-sbc 2990	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
7		df-sbc 2990	. 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 1474   e. wcel 2167   {cab 2182   [.wsbc 2989

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-sbc 2990

This theorem is referenced by:  cbvsbcv  3019  cbvcsb  3089