Theorem elab2 2989

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Hypotheses

Ref	Expression
elab2.1	|- A e. _V
elab2.2	|- (x = A -> (ph <-> ps))
elab2.3	|- B = {x | ph}

Assertion

Ref	Expression
elab2	|- (A e. B <-> ps)

Distinct variable groups:   ps,x   x,A

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

Proof of Theorem elab2

Step	Hyp	Ref	Expression
1		elab2.1	. 2 |- A e. _V
2		elab2.2	. . 3 |- (x = A -> (ph <-> ps))
3		elab2.3	. . 3 |- B = {x | ph}
4	2, 3	elab2g 2988	. 2 |- (A e. _V -> (A e. B <-> ps))
5	1, 4	ax-mp 5	1 |- (A e. B <-> ps)

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710  {cab 2339  _Vcvv 2860

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:  elpw  3729  elint  3933  opkelopkabg  4246  0ceven  4506  eventfin  4518  oddtfin  4519  dfphi2  4570  phi11lem1  4596  0cnelphi  4598  proj1op  4601  proj2op  4602  opabid  4696  oprabid  5551  elfuns  5830