Theorem cbvab 2222

Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)

Hypotheses

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

Assertion

Ref	Expression
cbvab	|- { x | ph } = { y | ps }

Proof of Theorem cbvab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvab.2	. . . . 5 |- F/ x ps
2	1	nfsb 1882	. . . 4 |- F/ x [ z / y ] ps
3		cbvab.1	. . . . . 6 |- F/ y ph
4		cbvab.3	. . . . . . . 8 |- ( x = y -> ( ph <-> ps ) )
5	4	equcoms 1652	. . . . . . 7 |- ( y = x -> ( ph <-> ps ) )
6	5	bicomd 140	. . . . . 6 |- ( y = x -> ( ps <-> ph ) )
7	3, 6	sbie 1732	. . . . 5 |- ( [ x / y ] ps <-> ph )
8		sbequ 1779	. . . . 5 |- ( x = z -> ( [ x / y ] ps <-> [ z / y ] ps ) )
9	7, 8	syl5bbr 193	. . . 4 |- ( x = z -> ( ph <-> [ z / y ] ps ) )
10	2, 9	sbie 1732	. . 3 |- ( [ z / x ] ph <-> [ z / y ] ps )
11		df-clab 2087	. . 3 |- ( z e. { x | ph } <-> [ z / x ] ph )
12		df-clab 2087	. . 3 |- ( z e. { y | ps } <-> [ z / y ] ps )
13	10, 11, 12	3bitr4i 211	. 2 |- ( z e. { x | ph } <-> z e. { y | ps } )
14	13	eqriv 2097	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1299   F/wnf 1404   e. wcel 1448   [wsb 1703   {cab 2086

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093

This theorem is referenced by:  cbvabv  2223  cbvrab  2639  cbvsbc  2889  cbvrabcsf  3015  dfdmf  4670  dfrnf  4718  funfvdm2f  5418  abrexex2g  5949  abrexex2  5953