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Mirrors > Home > ILE Home > Th. List > exanaliim | GIF version |
Description: A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.) |
Ref | Expression |
---|---|
exanaliim | ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | annimim 676 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓)) | |
2 | 1 | eximi 1587 | . 2 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ∃𝑥 ¬ (𝜑 → 𝜓)) |
3 | exnalim 1633 | . 2 ⊢ (∃𝑥 ¬ (𝜑 → 𝜓) → ¬ ∀𝑥(𝜑 → 𝜓)) | |
4 | 2, 3 | syl 14 | 1 ⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∀wal 1340 ∃wex 1479 |
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 604 ax-in2 605 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 |
This theorem is referenced by: rexnalim 2453 nssr 3197 nssssr 4194 brprcneu 5473 |
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