Theorem nfeld 2390

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 2227	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1576	. . 3 |- F/ y ph
3		nfcvd 2375	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2389	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2388	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1616	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1809	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1523	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   F/wnf 1508   E.wex 1540   e. wcel 2202   F/_wnfc 2361

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-cleq 2224  df-clel 2227  df-nfc 2363

This theorem is referenced by:  nfneld  2505  nfraldw  2564  nfraldxy  2565  nfrexdxy  2566  nfreudxy  2707  nfsbc1d  3048  nfsbcd  3051  sbcrext  3109  nfsbcdw  3161  nfbrd  4134  nfriotadxy  5979  nfixpxy  6885