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Mirrors > Home > ILE Home > Th. List > xornbidc | GIF version |
Description: Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
Ref | Expression |
---|---|
xornbidc | ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xor 1358 | . . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | |
2 | xor2dc 1372 | . . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))))) | |
3 | 2 | imp 123 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))) |
4 | 1, 3 | bitr4id 198 | . 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))) |
5 | 4 | ex 114 | 1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 DECID wdc 820 ⊻ wxo 1357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-xor 1358 |
This theorem is referenced by: xordc 1374 xordidc 1381 |
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