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Theorem axi5r 2582
 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 2137 . . 3 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hba1 2137 . . 3 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbim 2112 . 2 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓))
4 sp 2041 . . 3 (∀𝑥𝜓𝜓)
54imim2i 16 . 2 ((∀𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝜑𝜓))
63, 5alrimih 1741 1 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by: (None)
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