Theorem elab3g 3357

Description: Membership in a class abstraction, with a weaker antecedent than elabg 3351. (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 2764	. 2 |- F/_ x A
2		nfv 1843	. 2 |- F/ x ps
3		elab3g.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	elab3gf 3356	1 |- ( ( ps -> A e. B ) -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  elab3  3358  elssabg  4819  elrnmptg  5375  elrelimasn  5489  elmapg  7870  isust  22007  ellimc  23637  isismty  33600  clublem  37917