Theorem nfel 2401

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 2234	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2392	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2400	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2386	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1618	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1690	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1527	1 |- F/ x A e. B

Syntax hints:   /\ wa 104   = wceq 1402   F/wnf 1513   E.wex 1545   e. wcel 2209   F/_wnfc 2379

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381

This theorem is referenced by:  nfel1  2403  nfel2  2405  nfnel  2522  elabgf  2968  elrabf  2980  sbcel12g  3162  nfdisjv  4116  rabxfrd  4613  ffnfvf  5861  mptelixpg  7009  elabgft1  16723  elabgf2  16725  bj-rspgt  16731