Theorem nfel 2317

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 2161	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2308	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2316	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2302	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1553	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1625	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1462	1 |- F/ x A e. B

Syntax hints:   /\ wa 103   = wceq 1343   F/wnf 1448   E.wex 1480   e. wcel 2136   F/_wnfc 2295

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297

This theorem is referenced by:  nfel1  2319  nfel2  2321  nfnel  2438  elabgf  2868  elrabf  2880  sbcel12g  3060  nfdisjv  3971  rabxfrd  4447  ffnfvf  5644  mptelixpg  6700  elabgft1  13659  elabgf2  13661  bj-rspgt  13667