Theorem elabf 2985

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
elabf.1	|- F/xps
elabf.2	|- A e. _V
elabf.3	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
elabf	|- (A e. {x | ph} <-> ps)

Distinct variable group:   x,A

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

Proof of Theorem elabf

Step	Hyp	Ref	Expression
1		elabf.2	. 2 |- A e. _V
2		nfcv 2490	. . 3 |- F/_xA
3		elabf.1	. . 3 |- F/xps
4		elabf.3	. . 3 |- (x = A -> (ph <-> ps))
5	2, 3, 4	elabgf 2984	. 2 |- (A e. _V -> (A e. {x | ph} <-> ps))
6	1, 5	ax-mp 5	1 |- (A e. {x | ph} <-> ps)

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544   = 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:  elab  2986