Theorem nfeld 2190

Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfeqd.1	|- (ph -> F/_xA)
nfeqd.2	|- (ph -> F/_xB)

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 2033	. 2 |- (A e. B <-> E.y(y = A /\ y e. B))
2		nfv 1418	. . 3 |- F/yph
3		nfcvd 2176	. . . . 5 |- (ph -> F/_xy)
4		nfeqd.1	. . . . 5 |- (ph -> F/_xA)
5	3, 4	nfeqd 2189	. . . 4 |- (ph -> F/x y = A)
6		nfeqd.2	. . . . 5 |- (ph -> F/_xB)
7	6	nfcrd 2188	. . . 4 |- (ph -> F/x y e. B)
8	5, 7	nfand 1457	. . 3 |- (ph -> F/x(y = A /\ y e. B))
9	2, 8	nfexd 1641	. 2 |- (ph -> F/xE.y(y = A /\ y e. B))
10	1, 9	nfxfrd 1361	1 |- (ph -> F/x A e. B)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   F/wnf 1346  E.wex 1378   e. wcel 1390   F/_wnfc 2162

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164

This theorem is referenced by:  nfneld  2299  nfraldxy  2350  nfrexdxy  2351  nfreudxy  2477  nfsbc1d  2774  nfsbcd  2777  nfbrd  3798  nfriotadxy  5419