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Mirrors > Home > ILE Home > Th. List > exnalim | 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 1536 | . 2 ⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑) | |
2 | 1 | con2i 557 | 1 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1241 ∃wex 1381 |
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 544 ax-in2 545 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 |
This theorem is referenced by: exanaliim 1538 alexnim 1539 dtru 4284 brprcneu 5171 |
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