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Theorem bj-exalimsi 34553
Description: An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1974 proves. (Contributed by BJ, 29-Sep-2019.)
Hypotheses
Ref Expression
bj-exalimsi.1 (𝜑 → (𝜓𝜒))
bj-exalimsi.2 (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
Assertion
Ref Expression
bj-exalimsi (∃𝑥𝜑 → (∀𝑥𝜓𝜒))

Proof of Theorem bj-exalimsi
StepHypRef Expression
1 bj-exalimsi.2 . . 3 (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
21bj-exalims 34552 . 2 (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓𝜒)))
3 bj-exalimsi.1 . 2 (𝜑 → (𝜓𝜒))
42, 3mpg 1805 1 (∃𝑥𝜑 → (∀𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1541  wex 1787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817
This theorem depends on definitions:  df-bi 210  df-ex 1788
This theorem is referenced by: (None)
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