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Theorem nfan1 1456
 Description: A closed form of nfan 1457. (Contributed by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
nfan1.1 𝑥𝜑
nfan1.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfan1 𝑥(𝜑𝜓)

Proof of Theorem nfan1
StepHypRef Expression
1 nfan1.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
21nfrd 1413 . . . 4 (𝜑 → (𝜓 → ∀𝑥𝜓))
32imdistani 419 . . 3 ((𝜑𝜓) → (𝜑 ∧ ∀𝑥𝜓))
4 nfan1.1 . . . . 5 𝑥𝜑
54nfri 1412 . . . 4 (𝜑 → ∀𝑥𝜑)
6519.28h 1454 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
73, 6sylibr 137 . 2 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
87nfi 1351 1 𝑥(𝜑𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  Ⅎwnf 1349 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-gen 1338  ax-4 1400 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  nfan  1457  sbcralt  2834  sbcrext  2835  csbiebt  2886  riota5f  5492
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