Theorem elab3g 2931

Description: Membership in a class abstraction, with a weaker antecedent than elabg 2926. (Contributed by NM, 29-Aug-2006.)

Hypothesis

Ref	Expression
elab3g.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
elab3g	|- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )

Distinct variable groups:   ps, x   x, A

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem elab3g

Step	Hyp	Ref	Expression
1		nfcv 2350	. 2 |- F/_ x A
2		nfv 1552	. 2 |- F/ x ps
3		elab3g.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elab3gf 2930	1 |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178   {cab 2193

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:  elab3  2932  elssabg  4208  elrnmptg  4949  elrelimasn  5067  fvelrnb  5649  elmapg  6771  isghm  13694  ellspsn  14294