Theorem nfeld 2328

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 2166	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1521	. . 3 |- F/ y ph
3		nfcvd 2313	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2327	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2326	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1561	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1754	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1468	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   F/wnf 1453   E.wex 1485   e. wcel 2141   F/_wnfc 2299

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166  df-nfc 2301

This theorem is referenced by:  nfneld  2443  nfraldw  2502  nfraldxy  2503  nfrexdxy  2504  nfreudxy  2643  nfsbc1d  2971  nfsbcd  2974  sbcrext  3032  nfsbcdw  3083  nfbrd  4034  nfriotadxy  5817  nfixpxy  6695