Theorem nfel 2328

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 2173	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2319	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2327	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2313	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1565	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1637	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1474	1 |- F/ x A e. B

Syntax hints:   /\ wa 104   = wceq 1353   F/wnf 1460   E.wex 1492   e. wcel 2148   F/_wnfc 2306

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308

This theorem is referenced by:  nfel1  2330  nfel2  2332  nfnel  2449  elabgf  2881  elrabf  2893  sbcel12g  3074  nfdisjv  3994  rabxfrd  4471  ffnfvf  5677  mptelixpg  6736  elabgft1  14615  elabgf2  14617  bj-rspgt  14623