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Mirrors > Home > ILE Home > Th. List > xor2dc | GIF version |
Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.) |
Ref | Expression |
---|---|
xor2dc | ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xor3dc 1318 | . . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))) | |
2 | 1 | imp 122 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))) |
3 | pm5.17dc 843 | . . . 4 ⊢ (DECID 𝜓 → (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓))) | |
4 | 3 | adantl 271 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓))) |
5 | 2, 4 | bitr4d 189 | . 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))) |
6 | 5 | ex 113 | 1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 661 DECID wdc 775 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 df-dc 776 |
This theorem is referenced by: xornbidc 1322 |
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