MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axi5r Structured version   Visualization version   GIF version

Theorem axi5r 2623
Description: Converse of ax-c4 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)
Assertion
Ref Expression
axi5r ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑𝜓))

Proof of Theorem axi5r
StepHypRef Expression
1 hba1 2189 . . 3 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hba1 2189 . . 3 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbim 2165 . 2 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓))
4 sp 2091 . . 3 (∀𝑥𝜓𝜓)
54imim2i 16 . 2 ((∀𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝜑𝜓))
63, 5alrimih 1791 1 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator