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Theorem bj-alanim 32273
Description: Closed form of alanimi 1741. (Contributed by BJ, 6-May-2019.)
Assertion
Ref Expression
bj-alanim (∀𝑥((𝜑𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))

Proof of Theorem bj-alanim
StepHypRef Expression
1 pm3.3 460 . . . 4 (((𝜑𝜓) → 𝜒) → (𝜑 → (𝜓𝜒)))
21alimi 1736 . . 3 (∀𝑥((𝜑𝜓) → 𝜒) → ∀𝑥(𝜑 → (𝜓𝜒)))
3 al2im 1739 . . 3 (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
42, 3syl 17 . 2 (∀𝑥((𝜑𝜓) → 𝜒) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
54impd 447 1 (∀𝑥((𝜑𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386
This theorem is referenced by: (None)
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