Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-modal4e Structured version   Visualization version   GIF version

Theorem bj-modal4e 33558
 Description: Dual statement of hba1 2227 (which is modal-4 ). (Contributed by BJ, 21-Dec-2020.) TODO: in the proof, replace use of hba1 2227 by bj-hba1 33783 which is the standard proof. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-modal4e (∃𝑥𝑥𝜑 → ∃𝑥𝜑)

Proof of Theorem bj-modal4e
StepHypRef Expression
1 hba1 2227 . . 3 (∀𝑥 ¬ 𝜑 → ∀𝑥𝑥 ¬ 𝜑)
2 alnex 1744 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3 2exnaln 1791 . . . 4 (∃𝑥𝑥𝜑 ↔ ¬ ∀𝑥𝑥 ¬ 𝜑)
43con2bii 350 . . 3 (∀𝑥𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝑥𝜑)
51, 2, 43imtr3i 283 . 2 (¬ ∃𝑥𝜑 → ¬ ∃𝑥𝑥𝜑)
65con4i 114 1 (∃𝑥𝑥𝜑 → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1505  ∃wex 1742 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106 This theorem depends on definitions:  df-bi 199  df-or 834  df-ex 1743  df-nf 1747 This theorem is referenced by:  bj-nnfe1  33785
 Copyright terms: Public domain W3C validator