Theorem cbvabw 2289

Description: Version of cbvab 2290 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem cbvabw

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvabw.1	. . . . . 6 |- F/ y ph
2	1	nfsbv 1935	. . . . 5 |- F/ y [ z / x ] ph
3		equequ2 1701	. . . . . . . 8 |- ( y = z -> ( x = y <-> x = z ) )
4	3	imbi1d 230	. . . . . . 7 |- ( y = z -> ( ( x = y -> ph ) <-> ( x = z -> ph ) ) )
5	4	albidv 1812	. . . . . 6 |- ( y = z -> ( A. x ( x = y -> ph ) <-> A. x ( x = z -> ph ) ) )
6		sb6 1874	. . . . . 6 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
7		sb6 1874	. . . . . 6 |- ( [ z / x ] ph <-> A. x ( x = z -> ph ) )
8	5, 6, 7	3bitr4g 222	. . . . 5 |- ( y = z -> ( [ y / x ] ph <-> [ z / x ] ph ) )
9	2, 8	sbiev 1780	. . . 4 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
10		cbvabw.2	. . . . . 6 |- F/ x ps
11		cbvabw.3	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
12	10, 11	sbiev 1780	. . . . 5 |- ( [ y / x ] ph <-> ps )
13	12	sbbii 1753	. . . 4 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] ps )
14	9, 13	bitr3i 185	. . 3 |- ( [ z / x ] ph <-> [ z / y ] ps )
15		df-clab 2152	. . 3 |- ( z e. { x | ph } <-> [ z / x ] ph )
16		df-clab 2152	. . 3 |- ( z e. { y | ps } <-> [ z / y ] ps )
17	14, 15, 16	3bitr4i 211	. 2 |- ( z e. { x | ph } <-> z e. { y | ps } )
18	17	eqriv 2162	1 |- { x | ph } = { y | ps }

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   F/wnf 1448   [wsb 1750   e. wcel 2136   {cab 2151

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-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158

This theorem is referenced by:  cbvsbcw  2978