Theorem cbvrab 2555

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)

Hypotheses

Ref	Expression
cbvrab.1	|- F/_ x A
cbvrab.2	|- F/_ y A
cbvrab.3	|- F/ y ph
cbvrab.4	|- F/ x ps
cbvrab.5	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvrab	|- { x e. A | ph } = { y e. A | ps }

Proof of Theorem cbvrab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1421	. . . 4 |- F/ z ( x e. A /\ ph )
2		cbvrab.1	. . . . . 6 |- F/_ x A
3	2	nfcri 2172	. . . . 5 |- F/ x z e. A
4		nfs1v 1815	. . . . 5 |- F/ x [ z / x ] ph
5	3, 4	nfan 1457	. . . 4 |- F/ x ( z e. A /\ [ z / x ] ph )
6		eleq1 2100	. . . . 5 |- ( x = z -> ( x e. A <-> z e. A ) )
7		sbequ12 1654	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	6, 7	anbi12d 442	. . . 4 |- ( x = z -> ( ( x e. A /\ ph ) <-> ( z e. A /\ [ z / x ] ph ) ) )
9	1, 5, 8	cbvab 2160	. . 3 |- { x | ( x e. A /\ ph ) } = { z | ( z e. A /\ [ z / x ] ph ) }
10		cbvrab.2	. . . . . 6 |- F/_ y A
11	10	nfcri 2172	. . . . 5 |- F/ y z e. A
12		cbvrab.3	. . . . . 6 |- F/ y ph
13	12	nfsb 1822	. . . . 5 |- F/ y [ z / x ] ph
14	11, 13	nfan 1457	. . . 4 |- F/ y ( z e. A /\ [ z / x ] ph )
15		nfv 1421	. . . 4 |- F/ z ( y e. A /\ ps )
16		eleq1 2100	. . . . 5 |- ( z = y -> ( z e. A <-> y e. A ) )
17		sbequ 1721	. . . . . 6 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
18		cbvrab.4	. . . . . . 7 |- F/ x ps
19		cbvrab.5	. . . . . . 7 |- ( x = y -> ( ph <-> ps ) )
20	18, 19	sbie 1674	. . . . . 6 |- ( [ y / x ] ph <-> ps )
21	17, 20	syl6bb 185	. . . . 5 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
22	16, 21	anbi12d 442	. . . 4 |- ( z = y -> ( ( z e. A /\ [ z / x ] ph ) <-> ( y e. A /\ ps ) ) )
23	14, 15, 22	cbvab 2160	. . 3 |- { z | ( z e. A /\ [ z / x ] ph ) } = { y | ( y e. A /\ ps ) }
24	9, 23	eqtri 2060	. 2 |- { x | ( x e. A /\ ph ) } = { y | ( y e. A /\ ps ) }
25		df-rab 2315	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
26		df-rab 2315	. 2 |- { y e. A | ps } = { y | ( y e. A /\ ps ) }
27	24, 25, 26	3eqtr4i 2070	1 |- { x e. A | ph } = { y e. A | ps }

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   F/wnf 1349   e. wcel 1393   [wsb 1645   {cab 2026   F/_wnfc 2165   {crab 2310

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  df-clel 2036  df-nfc 2167  df-rab 2315

This theorem is referenced by:  cbvrabv  2556  elrabsf  2801  tfis  4306