Theorem nfel 2308

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 2153	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2299	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2307	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2293	. . . 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 1454	1 |- F/ x A e. B

Syntax hints:   /\ wa 103   = wceq 1335   F/wnf 1440   E.wex 1472   e. wcel 2128   F/_wnfc 2286

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288

This theorem is referenced by:  nfel1  2310  nfel2  2312  nfnel  2429  elabgf  2854  elrabf  2866  sbcel12g  3046  nfdisjv  3955  rabxfrd  4430  ffnfvf  5627  mptelixpg  6680  elabgft1  13394  elabgf2  13396  bj-rspgt  13402