Theorem cbvrab 1425

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.

Hypotheses

Ref	Expression
cbvrab.1	|- (z e. A -> A.x z e. A)
cbvrab.2	|- (z e. A -> A.y z e. A)
cbvrab.3	|- (ph -> A.yph)
cbvrab.4	|- (ps -> A.xps)
cbvrab.5	|- (x = y -> (ph <-> ps))

Assertion

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

Distinct variable group(s):   x,y,z   z,A

Proof of Theorem cbvrab

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 |- (z e. x -> A.y z e. x)
2		cbvrab.2	. . . . 5 |- (z e. A -> A.y z e. A)
3	1, 2	hbel 1172	. . . 4 |- (x e. A -> A.y x e. A)
4		cbvrab.3	. . . 4 |- (ph -> A.yph)
5	3, 4	hban 704	. . 3 |- ((x e. A /\ ph) -> A.y(x e. A /\ ph))
6		ax-17 925	. . . . 5 |- (z e. y -> A.x z e. y)
7		cbvrab.1	. . . . 5 |- (z e. A -> A.x z e. A)
8	6, 7	hbel 1172	. . . 4 |- (y e. A -> A.x y e. A)
9		cbvrab.4	. . . 4 |- (ps -> A.xps)
10	8, 9	hban 704	. . 3 |- ((y e. A /\ ps) -> A.x(y e. A /\ ps))
11		eleq1 1149	. . . 4 |- (x = y -> (x e. A <-> y e. A))
12		cbvrab.5	. . . 4 |- (x = y -> (ph <-> ps))
13	11, 12	anbi12d 476	. . 3 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ ps)))
14	5, 10, 13	cbvab 1423	. 2 |- {x | (x e. A /\ ph)} = {y | (y e. A /\ ps)}
15		df-rab 1208	. 2 |- {x e. A | ph} = {x | (x e. A /\ ph)}
16		df-rab 1208	. 2 |- {y e. A | ps} = {y | (y e. A /\ ps)}
17	14, 15, 16	3eqtr4 1126	1 |- {x e. A | ph} = {y e. A | ps}

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   = weq 797   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  {crab 1204

This theorem is referenced by:  cbvrabv 1426  elrabsf 1456  iunrab 2022  scottexs 3543  scott0s 3544  hta 3619

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208  df-v 1349

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