Theorem elab3g 2992

Description: Membership in a class abstraction, with a weaker antecedent than elabg 2987. (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 2490	. 2 |- F/_xA
2		nfv 1619	. 2 |- F/xps
3		elab3g.1	. 2 |- (x = A -> (ph <-> ps))
4	1, 2, 3	elab3gf 2991	1 |- ((ps -> A e. B) -> (A e. {x | ph} <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  elab3  2993