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Theorem eximal 1854
 Description: A utility theorem. 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 1905. (Contributed by BJ, 12-May-2019.)
Assertion
Ref Expression
eximal ((∃𝑥𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Proof of Theorem eximal
StepHypRef Expression
1 df-ex 1852 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
21imbi1i 338 . 2 ((∃𝑥𝜑𝜓) ↔ (¬ ∀𝑥 ¬ 𝜑𝜓))
3 con1b 347 . 2 ((¬ ∀𝑥 ¬ 𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
42, 3bitri 264 1 ((∃𝑥𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1628  ∃wex 1851 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 197  df-ex 1852 This theorem is referenced by:  ax5e  1992  19.23t  2234  19.23tOLD  2379  xfree2  29638  bj-exalims  32944
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