Description: A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | tsbi2 | ⊢ ( 𝜃 → ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.21 | ⊢ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) → ( 𝜑 ↔ 𝜓 ) ) | |
2 | 1 | olcd | ⊢ ( ( ¬ 𝜑 ∧ ¬ 𝜓 ) → ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) ) |
3 | pm4.57 | ⊢ ( ¬ ( ¬ 𝜑 ∧ ¬ 𝜓 ) ↔ ( 𝜑 ∨ 𝜓 ) ) | |
4 | 3 | biimpi | ⊢ ( ¬ ( ¬ 𝜑 ∧ ¬ 𝜓 ) → ( 𝜑 ∨ 𝜓 ) ) |
5 | 4 | orcd | ⊢ ( ¬ ( ¬ 𝜑 ∧ ¬ 𝜓 ) → ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) ) |
6 | 2 5 | pm2.61i | ⊢ ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) |
7 | 6 | a1i | ⊢ ( 𝜃 → ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ↔ 𝜓 ) ) ) |