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Theorem alimex 1832
 Description: An equivalence between an implication with a universally quantified consequent and an implication with an existentially quantified antecedent. An interesting case is when the same formula is substituted for both 𝜑 and 𝜓, since then both implications express a type of non-freeness. See also eximal 1784. (Contributed by BJ, 12-May-2019.)
Assertion
Ref Expression
alimex ((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))

Proof of Theorem alimex
StepHypRef Expression
1 alex 1827 . . 3 (∀𝑥𝜓 ↔ ¬ ∃𝑥 ¬ 𝜓)
21imbi2i 339 . 2 ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ¬ ∃𝑥 ¬ 𝜓))
3 con2b 363 . 2 ((𝜑 → ¬ ∃𝑥 ¬ 𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))
42, 3bitri 278 1 ((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀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-nnfnt  34465
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