Theorem cbvrab 2728

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 1521	. . . 4 |- F/ z ( x e. A /\ ph )
2		cbvrab.1	. . . . . 6 |- F/_ x A
3	2	nfcri 2306	. . . . 5 |- F/ x z e. A
4		nfs1v 1932	. . . . 5 |- F/ x [ z / x ] ph
5	3, 4	nfan 1558	. . . 4 |- F/ x ( z e. A /\ [ z / x ] ph )
6		eleq1 2233	. . . . 5 |- ( x = z -> ( x e. A <-> z e. A ) )
7		sbequ12 1764	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	6, 7	anbi12d 470	. . . 4 |- ( x = z -> ( ( x e. A /\ ph ) <-> ( z e. A /\ [ z / x ] ph ) ) )
9	1, 5, 8	cbvab 2294	. . 3 |- { x | ( x e. A /\ ph ) } = { z | ( z e. A /\ [ z / x ] ph ) }
10		cbvrab.2	. . . . . 6 |- F/_ y A
11	10	nfcri 2306	. . . . 5 |- F/ y z e. A
12		cbvrab.3	. . . . . 6 |- F/ y ph
13	12	nfsb 1939	. . . . 5 |- F/ y [ z / x ] ph
14	11, 13	nfan 1558	. . . 4 |- F/ y ( z e. A /\ [ z / x ] ph )
15		nfv 1521	. . . 4 |- F/ z ( y e. A /\ ps )
16		eleq1 2233	. . . . 5 |- ( z = y -> ( z e. A <-> y e. A ) )
17		sbequ 1833	. . . . . 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 1784	. . . . . 6 |- ( [ y / x ] ph <-> ps )
21	17, 20	bitrdi 195	. . . . 5 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
22	16, 21	anbi12d 470	. . . 4 |- ( z = y -> ( ( z e. A /\ [ z / x ] ph ) <-> ( y e. A /\ ps ) ) )
23	14, 15, 22	cbvab 2294	. . 3 |- { z | ( z e. A /\ [ z / x ] ph ) } = { y | ( y e. A /\ ps ) }
24	9, 23	eqtri 2191	. 2 |- { x | ( x e. A /\ ph ) } = { y | ( y e. A /\ ps ) }
25		df-rab 2457	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
26		df-rab 2457	. 2 |- { y e. A | ps } = { y | ( y e. A /\ ps ) }
27	24, 25, 26	3eqtr4i 2201	1 |- { x e. A | ph } = { y e. A | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   F/wnf 1453   [wsb 1755   e. wcel 2141   {cab 2156   F/_wnfc 2299   {crab 2452

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  df-clel 2166  df-nfc 2301  df-rab 2457

This theorem is referenced by:  cbvrabv  2729  elrabsf  2993  tfis  4567