Theorem elab3g 2977

Description: Membership in a class abstraction, with a weaker antecedent than elabg 2972. (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 2392	. 2 |- F/_ x A
2		nfv 1581	. 2 |- F/ x ps
3		elab3g.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elab3gf 2976	1 |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  elab3  2978  elssabg  4279  elrnmptg  5029  elrelimasn  5148  fvelrnb  5744  elmapg  6925  isghm  14023  ellspsn  14726