Theorem elab3gf 2930

Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2922. (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	. . . 4 |- F/_ x A
2		elab3gf.2	. . . 4 |- F/ x ps
3		elab3gf.3	. . . 4 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elabgf 2922	. . 3 |- ( A e. { x | ph } -> ( A e. { x | ph } <-> ps ) )
5	4	ibi 176	. 2 |- ( A e. { x | ph } -> ps )
6	1, 2, 3	elabgf 2922	. . . 4 |- ( A e. B -> ( A e. { x | ph } <-> ps ) )
7	6	imim2i 12	. . 3 |- ( ( ps -> A e. B ) -> ( ps -> ( A e. { x | ph } <-> ps ) ) )
8		biimpr 130	. . 3 |- ( ( A e. { x | ph } <-> ps ) -> ( ps -> A e. { x | ph } ) )
9	7, 8	syli 37	. 2 |- ( ( ps -> A e. B ) -> ( ps -> A e. { x | ph } ) )
10	5, 9	impbid2 143	1 |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   F/wnf 1484   e. wcel 2178   {cab 2193   F/_wnfc 2337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  elab3g  2931