Theorem cbvab 2294

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 1939	. . . 4 |- F/ x [ z / y ] ps
3		cbvab.1	. . . . . 6 |- F/ y ph
4		cbvab.3	. . . . . . . 8 |- ( x = y -> ( ph <-> ps ) )
5	4	equcoms 1701	. . . . . . 7 |- ( y = x -> ( ph <-> ps ) )
6	5	bicomd 140	. . . . . 6 |- ( y = x -> ( ps <-> ph ) )
7	3, 6	sbie 1784	. . . . 5 |- ( [ x / y ] ps <-> ph )
8		sbequ 1833	. . . . 5 |- ( x = z -> ( [ x / y ] ps <-> [ z / y ] ps ) )
9	7, 8	bitr3id 193	. . . 4 |- ( x = z -> ( ph <-> [ z / y ] ps ) )
10	2, 9	sbie 1784	. . 3 |- ( [ z / x ] ph <-> [ z / y ] ps )
11		df-clab 2157	. . 3 |- ( z e. { x | ph } <-> [ z / x ] ph )
12		df-clab 2157	. . 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 2167	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   F/wnf 1453   [wsb 1755   e. wcel 2141   {cab 2156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163

This theorem is referenced by:  cbvabv  2295  cbvrab  2728  cbvsbc  2983  cbvrabcsf  3114  dfdmf  4804  dfrnf  4852  funfvdm2f  5561  abrexex2g  6099  abrexex2  6103