Theorem elab3gf 2910

Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2902. (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 2902	. . 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 2902	. . . 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 1364   F/wnf 1471   e. wcel 2164   {cab 2179   F/_wnfc 2323

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762

This theorem is referenced by:  elab3g  2911