Theorem elrabf 2413

Description: Membership in a restricted class abstraction, using 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 groups:   y,A   y,B

Proof of Theorem elrabf

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (A e. {x e. B | ph} -> A e. _V)
2		elisset 2299	. . 3 |- (A e. B -> A e. _V)
3	2	adantr 425	. 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 1996	. . . . 5 |- (A e. B -> A.x A e. B)
7		elrabf.3	. . . . 5 |- (ps -> A.xps)
8	6, 7	hban 1356	. . . 4 |- ((A e. B /\ ps) -> A.x(A e. B /\ ps))
9		eleq1 1957	. . . . 5 |- (x = A -> (x e. B <-> A e. B))
10		elrabf.4	. . . . 5 |- (x = A -> (ph <-> ps))
11	9, 10	anbi12d 690	. . . 4 |- (x = A -> ((x e. B /\ ph) <-> (A e. B /\ ps)))
12	4, 8, 11	elabgf 2404	. . 3 |- (A e. _V -> (A e. {x | (x e. B /\ ph)} <-> (A e. B /\ ps)))
13		df-rab 2112	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
14	13	eleq2i 1961	. . 3 |- (A e. {x e. B | ph} <-> A e. {x | (x e. B /\ ph)})
15	12, 14	syl5bb 591	. 2 |- (A e. _V -> (A e. {x e. B | ph} <-> (A e. B /\ ps)))
16	1, 3, 15	pm5.21nii 743	1 |- (A e. {x e. B | ph} <-> (A e. B /\ ps))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871  {crab 2108  _Vcvv 2292

This theorem is referenced by:  elrab 2414  elrabsf 2486  rabxfrd 3842  onminsb 3879  tz9.12lem3 5772  ondomcard 6009  sltval2 13997  axfelem4 14034  fgsb 14921  fgsb2 14925  indexa 15753

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rab 2112  df-v 2294

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