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Theorem nfan1 1472
Description: A closed form of nfan 1473. (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 1429 . . . 4 (𝜑 → (𝜓 → ∀𝑥𝜓))
32imdistani 427 . . 3 ((𝜑𝜓) → (𝜑 ∧ ∀𝑥𝜓))
4 nfan1.1 . . . . 5 𝑥𝜑
54nfri 1428 . . . 4 (𝜑 → ∀𝑥𝜑)
6519.28h 1470 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
73, 6sylibr 141 . 2 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
87nfi 1367 1 𝑥(𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wnf 1365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  nfan  1473  sbcralt  2862  sbcrext  2863  csbiebt  2914  riota5f  5520
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