Theorem exp12bd

Description: The import-export theorem ( impexp ) for biconditionals (deduction form). (Contributed by Zhi Wang, 3-Sep-2024)

	Ref	Expression
Hypothesis	exp12bd.1	⊢ ( 𝜑 → ( ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( ( 𝜏 ∧ 𝜂 ) → 𝜁 ) ) )
Assertion	exp12bd	⊢ ( 𝜑 → ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) ↔ ( 𝜏 → ( 𝜂 → 𝜁 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		exp12bd.1	⊢ ( 𝜑 → ( ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( ( 𝜏 ∧ 𝜂 ) → 𝜁 ) ) )
2		impexp	⊢ ( ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) ↔ ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
3		impexp	⊢ ( ( ( 𝜏 ∧ 𝜂 ) → 𝜁 ) ↔ ( 𝜏 → ( 𝜂 → 𝜁 ) ) )
4	1 2 3	3bitr3g	⊢ ( 𝜑 → ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) ↔ ( 𝜏 → ( 𝜂 → 𝜁 ) ) ) )