Theorem elab3gf 3356

Description: Membership in a class abstraction, with a weaker antecedent than elabgf 3348. (Contributed by NM, 6-Sep-2011.)

Hypotheses

Ref	Expression
elab3gf.1	|- F/_ x A
elab3gf.2	|- F/ x ps
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/_ x A
2		elab3gf.2	. . . . 5 |- F/ x ps
3		elab3gf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabgf 3348	. . . 4 |- ( A e. { x | ph } -> ( A e. { x | ph } <-> ps ) )
5	4	ibi 256	. . 3 |- ( A e. { x | ph } -> ps )
6		pm2.21 120	. . 3 |- ( -. ps -> ( ps -> A e. { x | ph } ) )
7	5, 6	impbid2 216	. 2 |- ( -. ps -> ( A e. { x | ph } <-> ps ) )
8	1, 2, 3	elabgf 3348	. 2 |- ( A e. B -> ( A e. { x | ph } <-> ps ) )
9	7, 8	ja 173	1 |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   F/wnf 1708   e. wcel 1990   {cab 2608   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  elab3g  3357