Theorem elrabf 1421

Description: Membership in a restricted class abstraction with implicit substitution. This version has bound variable hypotheses in place of distinct variable restrictions.

Hypotheses

Ref	Expression
elrabf.1	|- (y e. A -> A.x y e. A)
elrabf.2	|- (y e. B -> A.x y e. B)
elrabf.3	|- (ps -> A.xps)
elrabf.4	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
elrabf	|- (A e. {x e. B | ph} <-> (A e. B /\ ps))

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

Proof of Theorem elrabf

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 |- (A e. {x e. B | ph} -> A e. V)
2		elisset 1354	. . 3 |- (A e. B -> A e. V)
3	2	adantr 306	. 2 |- ((A e. B /\ ps) -> A e. V)
4		elrabf.1	. . . 4 |- (y e. A -> A.x y e. A)
5		elrabf.2	. . . . . 6 |- (y e. B -> A.x y e. B)
6	4, 5	hbel 1172	. . . . 5 |- (A e. B -> A.x A e. B)
7		elrabf.3	. . . . 5 |- (ps -> A.xps)
8	6, 7	hban 704	. . . 4 |- ((A e. B /\ ps) -> A.x(A e. B /\ ps))
9		eleq1 1149	. . . . 5 |- (x = A -> (x e. B <-> A e. B))
10		elrabf.4	. . . . 5 |- (x = A -> (ph <-> ps))
11	9, 10	anbi12d 476	. . . 4 |- (x = A -> ((x e. B /\ ph) <-> (A e. B /\ ps)))
12	4, 8, 11	elabgf 1416	. . 3 |- (A e. V -> (A e. {x | (x e. B /\ ph)} <-> (A e. B /\ ps)))
13		df-rab 1208	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
14	13	eleq2i 1153	. . 3 |- (A e. {x e. B | ph} <-> A e. {x | (x e. B /\ ph)})
15	12, 14	syl5bb 410	. 2 |- (A e. V -> (A e. {x e. B | ph} <-> (A e. B /\ ps)))
16	1, 3, 15	pm5.21nii 504	1 |- (A e. {x e. B | ph} <-> (A e. B /\ ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  {cab 1090   = wceq 1091   e. wcel 1092  {crab 1204  Vcvv 1348

This theorem is referenced by:  elrab 1422  elrabsf 1456  onminsb 2264  tz9.12lem3 3505  ondomcard 3663

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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