Theorem cbvab 2160

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 1822	. . . 4 |- F/ x [ z / y ] ps
3		cbvab.1	. . . . . 6 |- F/ y ph
4		cbvab.3	. . . . . . . 8 |- ( x = y -> ( ph <-> ps ) )
5	4	equcoms 1594	. . . . . . 7 |- ( y = x -> ( ph <-> ps ) )
6	5	bicomd 129	. . . . . 6 |- ( y = x -> ( ps <-> ph ) )
7	3, 6	sbie 1674	. . . . 5 |- ( [ x / y ] ps <-> ph )
8		sbequ 1721	. . . . 5 |- ( x = z -> ( [ x / y ] ps <-> [ z / y ] ps ) )
9	7, 8	syl5bbr 183	. . . 4 |- ( x = z -> ( ph <-> [ z / y ] ps ) )
10	2, 9	sbie 1674	. . 3 |- ( [ z / x ] ph <-> [ z / y ] ps )
11		df-clab 2027	. . 3 |- ( z e. { x | ph } <-> [ z / x ] ph )
12		df-clab 2027	. . 3 |- ( z e. { y | ps } <-> [ z / y ] ps )
13	10, 11, 12	3bitr4i 201	. 2 |- ( z e. { x | ph } <-> z e. { y | ps } )
14	13	eqriv 2037	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   F/wnf 1349   e. wcel 1393   [wsb 1645   {cab 2026

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

This theorem is referenced by:  cbvabv  2161  cbvrab  2555  cbvsbc  2791  cbvrabcsf  2911  dfdmf  4528  dfrnf  4575  funfvdm2f  5238  abrexex2g  5747  abrexex2  5751