Theorem elrabf 1901

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 groups:   x,y   y,A   y,B

Proof of Theorem elrabf

Step	Hyp	Ref	Expression
1		elisset 1814	. 2 |- (A e. {x e. B | ph} -> A e. V)
2		elisset 1814	. . 3 |- (A e. B -> A e. V)
3	2	adantr 389	. 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 1564	. . . . 5 |- (A e. B -> A.x A e. B)
7		elrabf.3	. . . . 5 |- (ps -> A.xps)
8	6, 7	hban 1008	. . . 4 |- ((A e. B /\ ps) -> A.x(A e. B /\ ps))
9		eleq1 1532	. . . . 5 |- (x = A -> (x e. B <-> A e. B))
10		elrabf.4	. . . . 5 |- (x = A -> (ph <-> ps))
11	9, 10	anbi12d 627	. . . 4 |- (x = A -> ((x e. B /\ ph) <-> (A e. B /\ ps)))
12	4, 8, 11	elabgf 1895	. . 3 |- (A e. V -> (A e. {x | (x e. B /\ ph)} <-> (A e. B /\ ps)))
13		df-rab 1650	. . . 4 |- {x e. B | ph} = {x | (x e. B /\ ph)}
14	13	eleq2i 1536	. . 3 |- (A e. {x e. B | ph} <-> A e. {x | (x e. B /\ ph)})
15	12, 14	syl5bb 531	. 2 |- (A e. V -> (A e. {x e. B | ph} <-> (A e. B /\ ps)))
16	1, 3, 15	pm5.21nii 678	1 |- (A e. {x e. B | ph} <-> (A e. B /\ ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  {cab 1462  {crab 1646  Vcvv 1808

This theorem is referenced by:  elrab 1902  elrabsf 1960  rabxfr 2898  onminsb 3005  tz9.12lem3 4644  ondomcard 4840  fgsb 10503  fgsb2 10508

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-rab 1650  df-v 1809

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