Theorem elabf 2822

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 2279	. . 3 |- F/_ x A
3		elabf.1	. . 3 |- F/ x ps
4		elabf.3	. . 3 |- ( x = A -> ( ph <-> ps ) )
5	2, 3, 4	elabgf 2821	. 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 104   = wceq 1331   F/wnf 1436   e. wcel 1480   {cab 2123   _Vcvv 2681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  elab  2823  indpi  7143