Theorem nfel 2288

Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfnfc.1	|- F/_ x A
nfeq.2	|- F/_ x B

Assertion

Ref	Expression
nfel	|- F/ x A e. B

Proof of Theorem nfel

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clel 2133	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2279	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2287	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2273	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1544	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1616	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1450	1 |- F/ x A e. B

Syntax hints:   /\ wa 103   = wceq 1331   F/wnf 1436   E.wex 1468   e. wcel 1480   F/_wnfc 2266

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-cleq 2130  df-clel 2133  df-nfc 2268

This theorem is referenced by:  nfel1  2290  nfel2  2292  nfnel  2408  elabgf  2821  elrabf  2833  sbcel12g  3012  nfdisjv  3913  rabxfrd  4385  ffnfvf  5572  mptelixpg  6621  elabgft1  12974  elabgf2  12976  bj-rspgt  12982