Theorem elab3g 2911

Description: Membership in a class abstraction, with a weaker antecedent than elabg 2906. (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 2336	. 2 |- F/_ x A
2		nfv 1539	. 2 |- F/ x ps
3		elab3g.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elab3gf 2910	1 |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   {cab 2179

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:  elab3  2912  elssabg  4177  elrnmptg  4914  elrelimasn  5031  fvelrnb  5604  elmapg  6715  isghm  13313  ellspsn  13913