Theorem nfel 2357

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 2201	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2348	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2356	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2342	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1588	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1660	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1497	1 |- F/ x A e. B

Syntax hints:   /\ wa 104   = wceq 1373   F/wnf 1483   E.wex 1515   e. wcel 2176   F/_wnfc 2335

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337

This theorem is referenced by:  nfel1  2359  nfel2  2361  nfnel  2478  elabgf  2915  elrabf  2927  sbcel12g  3108  nfdisjv  4033  rabxfrd  4516  ffnfvf  5739  mptelixpg  6821  elabgft1  15714  elabgf2  15716  bj-rspgt  15722