Theorem nfel 2498

Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfnfc.1	|- F/_xA
nfeq.2	|- F/_xB

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 2349	. 2 |- (A e. B <-> E.z(z = A /\ z e. B))
2		nfcv 2490	. . . . 5 |- F/_xz
3		nfnfc.1	. . . . 5 |- F/_xA
4	2, 3	nfeq 2497	. . . 4 |- F/x z = A
5		nfeq.2	. . . . 5 |- F/_xB
6	5	nfcri 2484	. . . 4 |- F/x z e. B
7	4, 6	nfan 1824	. . 3 |- F/x(z = A /\ z e. B)
8	7	nfex 1843	. 2 |- F/xE.z(z = A /\ z e. B)
9	1, 8	nfxfr 1570	1 |- F/x A e. B

Syntax hints:   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by:  nfel1  2500  nfel2  2502  nfnel  2612  elabgf  2984  elrabf  2994  sbcel12g  3152  ffnfvf  5429