Theorem nfel 2186

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 2036	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2178	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2185	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2172	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1457	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1528	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1363	1 |- F/ x A e. B

Syntax hints:   /\ wa 97   = wceq 1243   F/wnf 1349   E.wex 1381   e. wcel 1393   F/_wnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167

This theorem is referenced by:  nfel1  2188  nfel2  2190  nfnel  2304  elabgf  2685  elrabf  2696  sbcel12g  2865  nfdisjv  3757  rabxfrd  4201  ffnfvf  5324  elabgft1  9917  elabgf2  9919  bj-rspgt  9925