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Theorem bj-exalims 34024
Description: Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1969 proves. (Contributed by BJ, 29-Sep-2019.)
Hypothesis
Ref Expression
bj-exalims.1 (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
Assertion
Ref Expression
bj-exalims (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓𝜒)))

Proof of Theorem bj-exalims
StepHypRef Expression
1 bj-exalim 34022 . 2 (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))
2 bj-exalims.1 . . . 4 (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
3 eximal 1784 . . . 4 ((∃𝑥𝜒𝜒) ↔ (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
42, 3sylibr 237 . . 3 (∃𝑥𝜑 → (∃𝑥𝜒𝜒))
54a1i 11 . 2 (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∃𝑥𝜒𝜒)))
61, 5syldd 72 1 (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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:  bj-exalimsi  34025
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