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Theorem axi5r 2784
Description: Converse of axc4 2340 (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 2301 . . 3 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hba1 2301 . . 3 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbim 2307 . 2 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓))
4 sp 2182 . . 3 (∀𝑥𝜓𝜓)
54imim2i 16 . 2 ((∀𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝜑𝜓))
63, 5alrimih 1824 1 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177
This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1781  df-nf 1785
This theorem is referenced by: (None)
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