ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfrimi GIF version

Theorem nfrimi 1525
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 1519 . . . 4 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
41nfri 1519 . . . 4 (𝜑 → ∀𝑥𝜑)
5 ax-5 1447 . . . 4 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
63, 4, 5syl2im 38 . . 3 ((𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
76pm2.86i 99 . 2 (𝜑 → (𝜓 → ∀𝑥𝜓))
81, 7nfd 1523 1 (𝜑 → Ⅎ𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351  wnf 1460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  hbsbd  1982
  Copyright terms: Public domain W3C validator