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Theorem nfeld 2190
 Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfeqd.1 (φxA)
nfeqd.2 (φxB)
Assertion
Ref Expression
nfeld (φ → Ⅎx A B)

Proof of Theorem nfeld
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-clel 2033 . 2 (A By(y = A y B))
2 nfv 1418 . . 3 yφ
3 nfcvd 2176 . . . . 5 (φxy)
4 nfeqd.1 . . . . 5 (φxA)
53, 4nfeqd 2189 . . . 4 (φ → Ⅎx y = A)
6 nfeqd.2 . . . . 5 (φxB)
76nfcrd 2188 . . . 4 (φ → Ⅎx y B)
85, 7nfand 1457 . . 3 (φ → Ⅎx(y = A y B))
92, 8nfexd 1641 . 2 (φ → Ⅎxy(y = A y B))
101, 9nfxfrd 1361 1 (φ → Ⅎx A B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  Ⅎwnf 1346  ∃wex 1378   ∈ wcel 1390  Ⅎwnfc 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
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