Theorem nfel 2383

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 2374	. . . . 5 |- F/_ x z
3		nfnfc.1	. . . . 5 |- F/_ x A
4	2, 3	nfeq 2382	. . . 4 |- F/ x z = A
5		nfeq.2	. . . . 5 |- F/_ x B
6	5	nfcri 2368	. . . 4 |- F/ x z e. B
7	4, 6	nfan 1613	. . 3 |- F/ x ( z = A /\ z e. B )
8	7	nfex 1685	. 2 |- F/ x E. z ( z = A /\ z e. B )
9	1, 8	nfxfr 1522	1 |- F/ x A e. B

Syntax hints:   /\ wa 104   = wceq 1397   F/wnf 1508   E.wex 1540   e. wcel 2202   F/_wnfc 2361

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2363

This theorem is referenced by:  nfel1  2385  nfel2  2387  nfnel  2504  elabgf  2948  elrabf  2960  sbcel12g  3142  nfdisjv  4076  rabxfrd  4566  ffnfvf  5806  mptelixpg  6902  elabgft1  16374  elabgf2  16376  bj-rspgt  16382