Theorem nfeld 2324

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 2161	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1516	. . 3 |- F/ y ph
3		nfcvd 2309	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2323	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2322	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1556	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1749	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1463	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   F/wnf 1448   E.wex 1480   e. wcel 2136   F/_wnfc 2295

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-cleq 2158  df-clel 2161  df-nfc 2297

This theorem is referenced by:  nfneld  2439  nfraldw  2498  nfraldxy  2499  nfrexdxy  2500  nfreudxy  2639  nfsbc1d  2967  nfsbcd  2970  sbcrext  3028  nfsbcdw  3079  nfbrd  4027  nfriotadxy  5806  nfixpxy  6683