Description: The connector \/_ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 21-Apr-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | xorcom | ⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( 𝜓 ⊻ 𝜑 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xor | ⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( 𝜑 ↔ 𝜓 ) ) | |
2 | bicom | ⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜓 ↔ 𝜑 ) ) | |
3 | 1 2 | xchbinx | ⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ¬ ( 𝜓 ↔ 𝜑 ) ) |
4 | df-xor | ⊢ ( ( 𝜓 ⊻ 𝜑 ) ↔ ¬ ( 𝜓 ↔ 𝜑 ) ) | |
5 | 3 4 | bitr4i | ⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( 𝜓 ⊻ 𝜑 ) ) |