Theorem elabf 3349

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/ x ps
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 2764	. . 3 |- F/_ x A
3		elabf.1	. . 3 |- F/ x ps
4		elabf.3	. . 3 |- ( x = A -> ( ph <-> ps ) )
5	2, 3, 4	elabgf 3348	. 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 196   = wceq 1483   F/wnf 1708   e. wcel 1990   {cab 2608   _Vcvv 3200

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:  elab  3350  dfon2lem1  31688  sdclem2  33538  sdclem1  33539