Theorem elab3gf 2991

Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2984. (Contributed by NM, 6-Sep-2011.)

Hypotheses

Ref	Expression
elab3gf.1	|- F/_xA
elab3gf.2	|- F/xps
elab3gf.3	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
elab3gf	|- ((ps -> A e. B) -> (A e. {x | ph} <-> ps))

Proof of Theorem elab3gf

Step	Hyp	Ref	Expression
1		elab3gf.1	. . . . 5 |- F/_xA
2		elab3gf.2	. . . . 5 |- F/xps
3		elab3gf.3	. . . . 5 |- (x = A -> (ph <-> ps))
4	1, 2, 3	elabgf 2984	. . . 4 |- (A e. {x | ph} -> (A e. {x | ph} <-> ps))
5	4	ibi 232	. . 3 |- (A e. {x | ph} -> ps)
6		pm2.21 100	. . 3 |- (-. ps -> (ps -> A e. {x | ph}))
7	5, 6	impbid2 195	. 2 |- (-. ps -> (A e. {x | ph} <-> ps))
8	1, 2, 3	elabgf 2984	. 2 |- (A e. B -> (A e. {x | ph} <-> ps))
9	7, 8	ja 153	1 |- ((ps -> A e. B) -> (A e. {x | ph} <-> ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   F/wnf 1544   = wceq 1642   e. wcel 1710  {cab 2339   F/_wnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  elab3g  2992