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Mirrors > Home > ILE Home > Th. List > pm4.63dc | GIF version |
Description: Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.) |
Ref | Expression |
---|---|
pm4.63dc | ⊢ (DECID φ → (DECID ψ → (¬ (φ → ¬ ψ) ↔ (φ ∧ ψ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfandc 777 | . . . 4 ⊢ (DECID φ → (DECID ψ → ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ)))) | |
2 | 1 | imp 115 | . . 3 ⊢ ((DECID φ ∧ DECID ψ) → ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ))) |
3 | 2 | bicomd 129 | . 2 ⊢ ((DECID φ ∧ DECID ψ) → (¬ (φ → ¬ ψ) ↔ (φ ∧ ψ))) |
4 | 3 | ex 108 | 1 ⊢ (DECID φ → (DECID ψ → (¬ (φ → ¬ ψ) ↔ (φ ∧ ψ)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 DECID wdc 741 |
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-io 629 |
This theorem depends on definitions: df-bi 110 df-dc 742 |
This theorem is referenced by: pm4.67dc 780 |
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