Theorem nfeld 2193

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 2036	. 2 |- ( A e. B <-> E. y ( y = A /\ y e. B ) )
2		nfv 1421	. . 3 |- F/ y ph
3		nfcvd 2179	. . . . 5 |- ( ph -> F/_ x y )
4		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
5	3, 4	nfeqd 2192	. . . 4 |- ( ph -> F/ x y = A )
6		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
7	6	nfcrd 2191	. . . 4 |- ( ph -> F/ x y e. B )
8	5, 7	nfand 1460	. . 3 |- ( ph -> F/ x ( y = A /\ y e. B ) )
9	2, 8	nfexd 1644	. 2 |- ( ph -> F/ x E. y ( y = A /\ y e. B ) )
10	1, 9	nfxfrd 1364	1 |- ( ph -> F/ x A e. B )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   F/wnf 1349   E.wex 1381   e. wcel 1393   F/_wnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167

This theorem is referenced by:  nfneld  2305  nfraldxy  2356  nfrexdxy  2357  nfreudxy  2483  nfsbc1d  2780  nfsbcd  2783  sbcrext  2835  nfbrd  3807  nfriotadxy  5476