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Theorem bj-exalimi 33986
Description: An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 33985 (using mpg 1799) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1967 proves. (Contributed by BJ, 29-Sep-2019.)
Hypothesis
Ref Expression
bj-exalimi.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
bj-exalimi (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))

Proof of Theorem bj-exalimi
StepHypRef Expression
1 bj-exalimi.1 . . . 4 (𝜑 → (𝜓𝜒))
21com12 32 . . 3 (𝜓 → (𝜑𝜒))
32aleximi 1833 . 2 (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥𝜒))
43com12 32 1 (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by: (None)
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