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Theorem alexim 1579
Description: One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1553. (Contributed by Jim Kingdon, 2-Jul-2018.)
Assertion
Ref Expression
alexim (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)

Proof of Theorem alexim
StepHypRef Expression
1 pm2.24 584 . . . . 5 (𝜑 → (¬ 𝜑 → ⊥))
21alimi 1387 . . . 4 (∀𝑥𝜑 → ∀𝑥𝜑 → ⊥))
3 exim 1533 . . . 4 (∀𝑥𝜑 → ⊥) → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥))
42, 3syl 14 . . 3 (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ∃𝑥⊥))
5 nfv 1464 . . . 4 𝑥
6519.9 1578 . . 3 (∃𝑥⊥ ↔ ⊥)
74, 6syl6ib 159 . 2 (∀𝑥𝜑 → (∃𝑥 ¬ 𝜑 → ⊥))
8 dfnot 1305 . 2 (¬ ∃𝑥 ¬ 𝜑 ↔ (∃𝑥 ¬ 𝜑 → ⊥))
97, 8sylibr 132 1 (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1285  wfal 1292  wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393
This theorem is referenced by:  exnalim  1580  exists2  2042
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