Theorem nfel 2238

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 2085	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2229	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2237	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2223	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1503	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1574	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1409	1 |- F/ x A e. B

Syntax hints:   /\ wa 103   = wceq 1290   F/wnf 1395   E.wex 1427   e. wcel 1439   F/_wnfc 2216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085  df-nfc 2218

This theorem is referenced by:  nfel1  2240  nfel2  2242  nfnel  2358  elabgf  2759  elrabf  2770  sbcel12g  2947  nfdisjv  3840  rabxfrd  4304  ffnfvf  5471  mptelixpg  6505  elabgft1  11951  elabgf2  11953  bj-rspgt  11959