Theorem cbvab 2210

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 1870	. . . 4 |- F/ x [ z / y ] ps
3		cbvab.1	. . . . . 6 |- F/ y ph
4		cbvab.3	. . . . . . . 8 |- ( x = y -> ( ph <-> ps ) )
5	4	equcoms 1641	. . . . . . 7 |- ( y = x -> ( ph <-> ps ) )
6	5	bicomd 139	. . . . . 6 |- ( y = x -> ( ps <-> ph ) )
7	3, 6	sbie 1721	. . . . 5 |- ( [ x / y ] ps <-> ph )
8		sbequ 1768	. . . . 5 |- ( x = z -> ( [ x / y ] ps <-> [ z / y ] ps ) )
9	7, 8	syl5bbr 192	. . . 4 |- ( x = z -> ( ph <-> [ z / y ] ps ) )
10	2, 9	sbie 1721	. . 3 |- ( [ z / x ] ph <-> [ z / y ] ps )
11		df-clab 2075	. . 3 |- ( z e. { x | ph } <-> [ z / x ] ph )
12		df-clab 2075	. . 3 |- ( z e. { y | ps } <-> [ z / y ] ps )
13	10, 11, 12	3bitr4i 210	. 2 |- ( z e. { x | ph } <-> z e. { y | ps } )
14	13	eqriv 2085	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   F/wnf 1394   e. wcel 1438   [wsb 1692   {cab 2074

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081

This theorem is referenced by:  cbvabv  2211  cbvrab  2617  cbvsbc  2865  cbvrabcsf  2991  dfdmf  4617  dfrnf  4664  funfvdm2f  5353  abrexex2g  5873  abrexex2  5877