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Theorem nfrimi 1461
Description: Moving an antecedent outside . (Contributed by Jim Kingdon, 23-Mar-2018.)
Hypotheses
Ref Expression
nfrimi.1 𝑥𝜑
nfrimi.2 𝑥(𝜑𝜓)
Assertion
Ref Expression
nfrimi (𝜑 → Ⅎ𝑥𝜓)

Proof of Theorem nfrimi
StepHypRef Expression
1 nfrimi.1 . 2 𝑥𝜑
2 nfrimi.2 . . . . 5 𝑥(𝜑𝜓)
32nfri 1455 . . . 4 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
41nfri 1455 . . . 4 (𝜑 → ∀𝑥𝜑)
5 ax-5 1379 . . . 4 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
63, 4, 5syl2im 38 . . 3 ((𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
76pm2.86i 97 . 2 (𝜑 → (𝜓 → ∀𝑥𝜓))
81, 7nfd 1459 1 (𝜑 → Ⅎ𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285  wnf 1392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-4 1443
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  hbsbd  1903
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