Theorem elabf 2882

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 2319	. . 3 |- F/_ x A
3		elabf.1	. . 3 |- F/ x ps
4		elabf.3	. . 3 |- ( x = A -> ( ph <-> ps ) )
5	2, 3, 4	elabgf 2881	. 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 105   = wceq 1353   F/wnf 1460   e. wcel 2148   {cab 2163   _Vcvv 2739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741

This theorem is referenced by:  elab  2883  indpi  7343