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Theorem alexim 1552
Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1526. (Contributed by Jim Kingdon, 2-Jul-2018.)
Assertion
Ref Expression
alexim (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)

Proof of Theorem alexim
StepHypRef Expression
1 pm2.24 561 . . . . 5 (𝜑 → (¬ 𝜑 → ⊥))
21alimi 1360 . . . 4 (∀𝑥𝜑 → ∀𝑥𝜑 → ⊥))
3 exim 1506 . . . 4 (∀𝑥𝜑 → ⊥) → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥))
42, 3syl 14 . . 3 (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥))
5 nfv 1437 . . . 4 𝑥
6519.9 1551 . . 3 (∃𝑥⊥ ↔ ⊥)
74, 6syl6ib 154 . 2 (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ⊥))
8 dfnot 1278 . 2 (¬ ∃𝑥 ¬ 𝜑 ↔ (∃𝑥 ¬ 𝜑 → ⊥))
97, 8sylibr 141 1 (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1257  wfal 1264  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366
This theorem is referenced by:  exnalim  1553  exists2  2013
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