Theorem nfeld 2297

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 2135	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1508	. . 3 |- F/ y ph
3		nfcvd 2282	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2296	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2295	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1547	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1734	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1451	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   F/wnf 1436   E.wex 1468   e. wcel 1480   F/_wnfc 2268

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2132  df-clel 2135  df-nfc 2270

This theorem is referenced by:  nfneld  2411  nfraldxy  2467  nfrexdxy  2468  nfreudxy  2604  nfsbc1d  2925  nfsbcd  2928  sbcrext  2986  nfbrd  3973  nfriotadxy  5738  nfixpxy  6611