Description: This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010)
Ref | Expression | ||
---|---|---|---|
Assertion | excxor | ⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( ( 𝜑 ∧ ¬ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xor | ⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( 𝜑 ↔ 𝜓 ) ) | |
2 | xor | ⊢ ( ¬ ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜑 ∧ ¬ 𝜓 ) ∨ ( 𝜓 ∧ ¬ 𝜑 ) ) ) | |
3 | ancom | ⊢ ( ( 𝜓 ∧ ¬ 𝜑 ) ↔ ( ¬ 𝜑 ∧ 𝜓 ) ) | |
4 | 3 | orbi2i | ⊢ ( ( ( 𝜑 ∧ ¬ 𝜓 ) ∨ ( 𝜓 ∧ ¬ 𝜑 ) ) ↔ ( ( 𝜑 ∧ ¬ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ) |
5 | 1 2 4 | 3bitri | ⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( ( 𝜑 ∧ ¬ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ) |