Theorem nfel 2359

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 2203	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2350	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2358	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2344	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1589	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1661	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1498	1 |- F/ x A e. B

Syntax hints:   /\ wa 104   = wceq 1373   F/wnf 1484   E.wex 1516   e. wcel 2178   F/_wnfc 2337

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-cleq 2200  df-clel 2203  df-nfc 2339

This theorem is referenced by:  nfel1  2361  nfel2  2363  nfnel  2480  elabgf  2922  elrabf  2934  sbcel12g  3116  nfdisjv  4047  rabxfrd  4534  ffnfvf  5762  mptelixpg  6844  elabgft1  15914  elabgf2  15916  bj-rspgt  15922