Theorem nfel 2345

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 2189	. 2 |- ( A e. B <-> E. z ( z = A /\ z e. B ) )
2		nfcv 2336	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2344	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2330	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1576	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1648	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1485	1 |- F/ x A e. B

Syntax hints:   /\ wa 104   = wceq 1364   F/wnf 1471   E.wex 1503   e. wcel 2164   F/_wnfc 2323

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2186  df-clel 2189  df-nfc 2325

This theorem is referenced by:  nfel1  2347  nfel2  2349  nfnel  2466  elabgf  2902  elrabf  2914  sbcel12g  3095  nfdisjv  4018  rabxfrd  4500  ffnfvf  5717  mptelixpg  6788  elabgft1  15270  elabgf2  15272  bj-rspgt  15278