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Theorem axi5r 2780
Description: Converse of axc4 2331 (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 2292 . . 3 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
2 hba1 2292 . . 3 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
31, 2hbim 2298 . 2 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓))
4 sp 2172 . . 3 (∀𝑥𝜓𝜓)
54imim2i 16 . 2 ((∀𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝜑𝜓))
63, 5alrimih 1815 1 ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-or 842  df-ex 1772  df-nf 1776
This theorem is referenced by: (None)
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