Theorem nfel 2291

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 2136	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2282	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2290	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2276	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1545	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1617	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1451	1 |- F/ x A e. B

Syntax hints:   /\ wa 103   = wceq 1332   F/wnf 1437   E.wex 1469   e. wcel 1481   F/_wnfc 2269

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271

This theorem is referenced by:  nfel1  2293  nfel2  2295  nfnel  2411  elabgf  2830  elrabf  2842  sbcel12g  3022  nfdisjv  3926  rabxfrd  4398  ffnfvf  5587  mptelixpg  6636  elabgft1  13156  elabgf2  13158  bj-rspgt  13164