Theorem nfeld 2345

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 2183	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1538	. . 3 |- F/ y ph
3		nfcvd 2330	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2344	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2343	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1578	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1771	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1485	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   F/wnf 1470   E.wex 1502   e. wcel 2158   F/_wnfc 2316

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-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-cleq 2180  df-clel 2183  df-nfc 2318

This theorem is referenced by:  nfneld  2460  nfraldw  2519  nfraldxy  2520  nfrexdxy  2521  nfreudxy  2661  nfsbc1d  2991  nfsbcd  2994  sbcrext  3052  nfsbcdw  3103  nfbrd  4060  nfriotadxy  5852  nfixpxy  6731