Theorem nfel 2384

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 2227	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2375	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2383	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2369	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1614	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1686	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1523	1 |- F/ x A e. B

Syntax hints:   /\ wa 104   = wceq 1398   F/wnf 1509   E.wex 1541   e. wcel 2202   F/_wnfc 2362

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2364

This theorem is referenced by:  nfel1  2386  nfel2  2388  nfnel  2505  elabgf  2949  elrabf  2961  sbcel12g  3143  nfdisjv  4081  rabxfrd  4572  ffnfvf  5814  mptelixpg  6946  elabgft1  16496  elabgf2  16498  bj-rspgt  16504