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Theorem eximal 1878
Description: An equivalence between an implication with an existentially quantified antecedent and an implication with a universally quantified consequent. 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 alimex 1926. (Contributed by BJ, 12-May-2019.)
Assertion
Ref Expression
eximal ((∃𝑥𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Proof of Theorem eximal
StepHypRef Expression
1 df-ex 1876 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
21imbi1i 341 . 2 ((∃𝑥𝜑𝜓) ↔ (¬ ∀𝑥 ¬ 𝜑𝜓))
3 con1b 350 . 2 ((¬ ∀𝑥 ¬ 𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
42, 3bitri 267 1 ((∃𝑥𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wal 1651  wex 1875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-ex 1876
This theorem is referenced by:  ax5e  2008  19.23tOLD  2245  xfree2  29829  bj-exalims  33118
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