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Theorem nfeld 2504
 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 2349 . 2 (A By(y = A y B))
2 nfv 1619 . . 3 yφ
3 nfcvd 2490 . . . . 5 (φxy)
4 nfeqd.1 . . . . 5 (φxA)
53, 4nfeqd 2503 . . . 4 (φ → Ⅎx y = A)
6 nfeqd.2 . . . . 5 (φxB)
76nfcrd 2502 . . . 4 (φ → Ⅎx y B)
85, 7nfand 1822 . . 3 (φ → Ⅎx(y = A y B))
92, 8nfexd 1854 . 2 (φ → Ⅎxy(y = A y B))
101, 9nfxfrd 1571 1 (φ → Ⅎx A B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-nfc 2478 This theorem is referenced by:  nfneld  2613  nfrald  2665  ralcom2  2775  nfreud  2783  nfrmod  2784  nfrmo  2786  nfsbc1d  3063  nfsbcd  3066  sbcrext  3119  nfbrd  4682
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