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Mirrors > Home > ILE Home > Th. List > annimim | GIF version |
Description: Express conjunction in terms of implication. One direction of Theorem *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as can be seen at annimdc 844. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Ref | Expression |
---|---|
annimim | ⊢ ((φ ∧ ¬ ψ) → ¬ (φ → ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.27 35 | . . 3 ⊢ (φ → ((φ → ψ) → ψ)) | |
2 | con3 570 | . . 3 ⊢ (((φ → ψ) → ψ) → (¬ ψ → ¬ (φ → ψ))) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (φ → (¬ ψ → ¬ (φ → ψ))) |
4 | 3 | imp 115 | 1 ⊢ ((φ ∧ ¬ ψ) → ¬ (φ → ψ)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-in1 544 ax-in2 545 |
This theorem is referenced by: pm4.65r 782 dcim 783 imanim 784 pm4.52im 802 exanaliim 1535 |
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