Theorem elab3g 1905

Description: Membership in a class abstraction, with a weaker antecedent than elabg 1902.

Hypothesis

Ref	Expression
elab3g.1	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
elab3g	|- ((ps -> A e. B) -> (A e. {x | ph} <-> ps))

Distinct variable groups:   ps,x   x,A

Proof of Theorem elab3g

Step	Hyp	Ref	Expression
1		elab3g.1	. . . 4 |- (x = A -> (ph <-> ps))
2	1	elabg 1902	. . 3 |- (A e. {x | ph} -> (A e. {x | ph} <-> ps))
3	2	ibi 594	. 2 |- (A e. {x | ph} -> ps)
4	1	elabg 1902	. . . 4 |- (A e. B -> (A e. {x | ph} <-> ps))
5	4	imim2i 17	. . 3 |- ((ps -> A e. B) -> (ps -> (A e. {x | ph} <-> ps)))
6		ibibr 593	. . 3 |- ((ps -> A e. {x | ph}) <-> (ps -> (A e. {x | ph} <-> ps)))
7	5, 6	sylibr 200	. 2 |- ((ps -> A e. B) -> (ps -> A e. {x | ph}))
8	3, 7	impbid2 520	1 |- ((ps -> A e. B) -> (A e. {x | ph} <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  {cab 1466

This theorem is referenced by:  elab3 1906  elssabg 2731  elmapg 4339  isnei 7715  ishomc 10688

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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