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Theorem al3im 36850
Description: Version of ax-4 1713 for a nested implication. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
al3im (∀𝑥(𝜑 → (𝜓 → (𝜒𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃))))

Proof of Theorem al3im
StepHypRef Expression
1 alim 1714 . 2 (∀𝑥(𝜑 → (𝜓 → (𝜒𝜃))) → (∀𝑥𝜑 → ∀𝑥(𝜓 → (𝜒𝜃))))
2 al2im 1717 . 2 (∀𝑥(𝜓 → (𝜒𝜃)) → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃)))
31, 2syl6 34 1 (∀𝑥(𝜑 → (𝜓 → (𝜒𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-4 1713
This theorem is referenced by: (None)
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