Description: A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | tsbi4 | ⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsbi3 | ⊢ ( 𝜃 → ( ( 𝜓 ∨ ¬ 𝜑 ) ∨ ¬ ( 𝜓 ↔ 𝜑 ) ) ) | |
2 | orcom | ⊢ ( ( 𝜓 ∨ ¬ 𝜑 ) ↔ ( ¬ 𝜑 ∨ 𝜓 ) ) | |
3 | bicom | ⊢ ( ( 𝜓 ↔ 𝜑 ) ↔ ( 𝜑 ↔ 𝜓 ) ) | |
4 | 3 | notbii | ⊢ ( ¬ ( 𝜓 ↔ 𝜑 ) ↔ ¬ ( 𝜑 ↔ 𝜓 ) ) |
5 | 2 4 | orbi12i | ⊢ ( ( ( 𝜓 ∨ ¬ 𝜑 ) ∨ ¬ ( 𝜓 ↔ 𝜑 ) ) ↔ ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) ) |
6 | 1 5 | sylib | ⊢ ( 𝜃 → ( ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) ) |