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

Theorem hbnd 1633
 Description: Deduction form of bound-variable hypothesis builder hbn 1632. (Contributed by NM, 3-Jan-2002.)
Hypotheses
Ref Expression
hbnd.1 (𝜑 → ∀𝑥𝜑)
hbnd.2 (𝜑 → (𝜓 → ∀𝑥𝜓))
Assertion
Ref Expression
hbnd (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))

Proof of Theorem hbnd
StepHypRef Expression
1 hbnd.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 hbnd.2 . . 3 (𝜑 → (𝜓 → ∀𝑥𝜓))
31, 2alrimih 1445 . 2 (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
4 hbnt 1631 . 2 (∀𝑥(𝜓 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
53, 4syl 14 1 (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-4 1487  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator