MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alimex Structured version   Visualization version   GIF version

Theorem alimex 1798
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 eximal 1747. (Contributed by BJ, 12-May-2019.)
Assertion
Ref Expression
alimex ((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))

Proof of Theorem alimex
StepHypRef Expression
1 alex 1793 . . 3 (∀𝑥𝜓 ↔ ¬ ∃𝑥 ¬ 𝜓)
21imbi2i 325 . 2 ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ¬ ∃𝑥 ¬ 𝜓))
3 con2b 348 . 2 ((𝜑 → ¬ ∃𝑥 ¬ 𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))
42, 3bitri 264 1 ((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator