Theorem elab3gf 2803

Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2796. (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 2796	. . 3 |- ( A e. { x | ph } -> ( A e. { x | ph } <-> ps ) )
5	4	ibi 175	. 2 |- ( A e. { x | ph } -> ps )
6	1, 2, 3	elabgf 2796	. . . 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		bi2 129	. . 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 142	1 |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1314   F/wnf 1419   e. wcel 1463   {cab 2101   F/_wnfc 2242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659

This theorem is referenced by:  elab3g  2804