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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 781 | . . 3 ⊢ ((φ ∧ ¬ ψ) → ¬ (φ → ψ)) | |
2 | 1 | eximi 1488 | . 2 ⊢ (∃x(φ ∧ ¬ ψ) → ∃x ¬ (φ → ψ)) |
3 | exnalim 1534 | . 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 97 ∀wal 1240 ∃wex 1378 |
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 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-fal 1248 df-nf 1347 |
This theorem is referenced by: rexnalim 2311 nssr 2997 nssssr 3949 brprcneu 5114 |
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