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Mirrors > Home > ILE Home > Th. List > exnalim | Structured version GIF version |
Description: One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.) |
Ref | Expression |
---|---|
exnalim | ⊢ (∃x ¬ φ → ¬ ∀xφ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alexim 1520 | . 2 ⊢ (∀xφ → ¬ ∃x ¬ φ) | |
2 | 1 | con2i 545 | 1 ⊢ (∃x ¬ φ → ¬ ∀xφ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1226 ∃wex 1363 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 532 ax-in2 533 ax-5 1316 ax-gen 1318 ax-ie1 1364 ax-ie2 1365 ax-4 1382 ax-17 1401 ax-ial 1410 |
This theorem depends on definitions: df-bi 110 df-tru 1231 df-fal 1234 df-nf 1330 |
This theorem is referenced by: exanaliim 1522 alexnim 1523 dtru 4220 brprcneu 5094 |
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