Description: A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | tsbi3 | ⊢ ( 𝜃 → ( ( 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimpr | ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) ) | |
2 | con34b | ⊢ ( ( 𝜓 → 𝜑 ) ↔ ( ¬ 𝜑 → ¬ 𝜓 ) ) | |
3 | pm2.54 | ⊢ ( ( ¬ 𝜑 → ¬ 𝜓 ) → ( 𝜑 ∨ ¬ 𝜓 ) ) | |
4 | 2 3 | sylbi | ⊢ ( ( 𝜓 → 𝜑 ) → ( 𝜑 ∨ ¬ 𝜓 ) ) |
5 | 1 4 | syl | ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 ∨ ¬ 𝜓 ) ) |
6 | 5 | con3i | ⊢ ( ¬ ( 𝜑 ∨ ¬ 𝜓 ) → ¬ ( 𝜑 ↔ 𝜓 ) ) |
7 | 6 | orri | ⊢ ( ( 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) |
8 | 7 | a1i | ⊢ ( 𝜃 → ( ( 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ ( 𝜑 ↔ 𝜓 ) ) ) |