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Mirrors > Home > ILE Home > Th. List > pm5.24dc | GIF version |
Description: Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
Ref | Expression |
---|---|
pm5.24dc | ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfbi3dc 1340 | . . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))))) | |
2 | 1 | imp 123 | . . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))) |
3 | 2 | notbid 630 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))) |
4 | xordc 1335 | . . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))) | |
5 | 4 | imp 123 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))) |
6 | 3, 5 | bitr3d 189 | . 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))) |
7 | 6 | ex 114 | 1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 667 DECID wdc 783 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-xor 1319 |
This theorem is referenced by: (None) |
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