Theorem nfeld 2364

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 2201	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1551	. . 3 |- F/ y ph
3		nfcvd 2349	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2363	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2362	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1591	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1784	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1498	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   F/wnf 1483   E.wex 1515   e. wcel 2176   F/_wnfc 2335

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-cleq 2198  df-clel 2201  df-nfc 2337

This theorem is referenced by:  nfneld  2479  nfraldw  2538  nfraldxy  2539  nfrexdxy  2540  nfreudxy  2680  nfsbc1d  3015  nfsbcd  3018  sbcrext  3076  nfsbcdw  3127  nfbrd  4089  nfriotadxy  5908  nfixpxy  6804