Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-exalimi Structured version   Visualization version   GIF version

Theorem bj-exalimi 31801
 Description: An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 1865 proves. (Contributed by BJ, 29-Sep-2019.)
Hypotheses
Ref Expression
bj-exalimi.1 (𝜑 → (𝜓𝜒))
bj-exalimi.2 (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
Assertion
Ref Expression
bj-exalimi (∃𝑥𝜑 → (∀𝑥𝜓𝜒))

Proof of Theorem bj-exalimi
StepHypRef Expression
1 bj-exalimi.1 . . 3 (𝜑 → (𝜓𝜒))
21bj-exaleximi 31800 . 2 (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))
3 bj-exalimi.2 . . 3 (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
4 eximal 1698 . . 3 ((∃𝑥𝜒𝜒) ↔ (¬ 𝜒 → ∀𝑥 ¬ 𝜒))
53, 4sylibr 223 . 2 (∃𝑥𝜑 → (∃𝑥𝜒𝜒))
62, 5syld 46 1 (∃𝑥𝜑 → (∀𝑥𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator