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Theorem alexim 1624
 Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1598. (Contributed by Jim Kingdon, 2-Jul-2018.)
Assertion
Ref Expression
alexim (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)

Proof of Theorem alexim
StepHypRef Expression
1 pm2.24 610 . . . . 5 (𝜑 → (¬ 𝜑 → ⊥))
21alimi 1431 . . . 4 (∀𝑥𝜑 → ∀𝑥𝜑 → ⊥))
3 exim 1578 . . . 4 (∀𝑥𝜑 → ⊥) → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥))
42, 3syl 14 . . 3 (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥))
5 nfv 1508 . . . 4 𝑥
6519.9 1623 . . 3 (∃𝑥⊥ ↔ ⊥)
74, 6syl6ib 160 . 2 (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ⊥))
8 dfnot 1349 . 2 (¬ ∃𝑥 ¬ 𝜑 ↔ (∃𝑥 ¬ 𝜑 → ⊥))
97, 8sylibr 133 1 (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1329  ⊥wfal 1336  ∃wex 1468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437 This theorem is referenced by:  exnalim  1625  exists2  2096
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