Theorem nfeld 2315

Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfeqd.1	|- ( ph -> F/_ x A )
nfeqd.2	|- ( ph -> F/_ x B )

Assertion

Ref	Expression
nfeld	|- ( ph -> F/ x A e. B )

Proof of Theorem nfeld

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clel 2153	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1508	. . 3 |- F/ y ph
3		nfcvd 2300	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2314	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2313	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1548	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1741	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1455	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   F/wnf 1440   E.wex 1472   e. wcel 2128   F/_wnfc 2286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-cleq 2150  df-clel 2153  df-nfc 2288

This theorem is referenced by:  nfneld  2430  nfraldw  2489  nfraldxy  2490  nfrexdxy  2491  nfreudxy  2630  nfsbc1d  2953  nfsbcd  2956  sbcrext  3014  nfsbcdw  3065  nfbrd  4009  nfriotadxy  5788  nfixpxy  6662