Theorem nfeld 2234

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 2077	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1461	. . 3 |- F/ y ph
3		nfcvd 2220	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2233	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2232	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1500	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1684	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1404	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   F/wnf 1389   E.wex 1421   e. wcel 1433   F/_wnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077  df-nfc 2208

This theorem is referenced by:  nfneld  2347  nfraldxy  2398  nfrexdxy  2399  nfreudxy  2527  nfsbc1d  2831  nfsbcd  2834  sbcrext  2891  nfbrd  3828  nfriotadxy  5496