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Mirrors > Home > MPE Home > Th. List > xor | Structured version Visualization version GIF version |
Description: Two ways to express "exclusive or." Theorem *5.22 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 22-Jan-2013.) |
Ref | Expression |
---|---|
xor | ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iman 439 | . . . 4 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) | |
2 | iman 439 | . . . 4 ⊢ ((𝜓 → 𝜑) ↔ ¬ (𝜓 ∧ ¬ 𝜑)) | |
3 | 1, 2 | anbi12i 729 | . . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (¬ (𝜑 ∧ ¬ 𝜓) ∧ ¬ (𝜓 ∧ ¬ 𝜑))) |
4 | dfbi2 658 | . . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) | |
5 | ioran 510 | . . 3 ⊢ (¬ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)) ↔ (¬ (𝜑 ∧ ¬ 𝜓) ∧ ¬ (𝜓 ∧ ¬ 𝜑))) | |
6 | 3, 4, 5 | 3bitr4ri 292 | . 2 ⊢ (¬ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)) ↔ (𝜑 ↔ 𝜓)) |
7 | 6 | con1bii 345 | 1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 |
This theorem is referenced by: dfbi3 933 pm5.24 934 4exmid 977 excxor 1461 elsymdif 3811 symdif2 3814 rpnnen2lem12 14793 ist0-3 20959 eliuniincex 38323 eliincex 38324 abnotataxb 39732 ldepslinc 42092 |
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