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Mirrors > Home > ILE Home > Th. List > pm5.15dc | GIF version |
Description: A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
Ref | Expression |
---|---|
pm5.15dc | ⊢ (DECID φ → (DECID ψ → ((φ ↔ ψ) ∨ (φ ↔ ¬ ψ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xor3dc 1275 | . . . . 5 ⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ (φ ↔ ¬ ψ)))) | |
2 | 1 | imp 115 | . . . 4 ⊢ ((DECID φ ∧ DECID ψ) → (¬ (φ ↔ ψ) ↔ (φ ↔ ¬ ψ))) |
3 | 2 | biimpd 132 | . . 3 ⊢ ((DECID φ ∧ DECID ψ) → (¬ (φ ↔ ψ) → (φ ↔ ¬ ψ))) |
4 | dcbi 843 | . . . . 5 ⊢ (DECID φ → (DECID ψ → DECID (φ ↔ ψ))) | |
5 | 4 | imp 115 | . . . 4 ⊢ ((DECID φ ∧ DECID ψ) → DECID (φ ↔ ψ)) |
6 | dfordc 790 | . . . 4 ⊢ (DECID (φ ↔ ψ) → (((φ ↔ ψ) ∨ (φ ↔ ¬ ψ)) ↔ (¬ (φ ↔ ψ) → (φ ↔ ¬ ψ)))) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ ((DECID φ ∧ DECID ψ) → (((φ ↔ ψ) ∨ (φ ↔ ¬ ψ)) ↔ (¬ (φ ↔ ψ) → (φ ↔ ¬ ψ)))) |
8 | 3, 7 | mpbird 156 | . 2 ⊢ ((DECID φ ∧ DECID ψ) → ((φ ↔ ψ) ∨ (φ ↔ ¬ ψ))) |
9 | 8 | 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 ∨ wo 628 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: (None) |
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