Theorem nfeld 2352

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 2189	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1539	. . 3 |- F/ y ph
3		nfcvd 2337	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2351	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2350	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1579	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1772	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1486	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   F/wnf 1471   E.wex 1503   e. wcel 2164   F/_wnfc 2323

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-cleq 2186  df-clel 2189  df-nfc 2325

This theorem is referenced by:  nfneld  2467  nfraldw  2526  nfraldxy  2527  nfrexdxy  2528  nfreudxy  2668  nfsbc1d  3002  nfsbcd  3005  sbcrext  3063  nfsbcdw  3114  nfbrd  4074  nfriotadxy  5882  nfixpxy  6771