Theorem exp12bd 46141

Description: The import-export theorem (impexp 451) for biconditionals (deduction form). (Contributed by Zhi Wang, 3-Sep-2024.)

Hypothesis

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

Assertion

Ref	Expression
exp12bd	⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) ↔ (𝜏 → (𝜂 → 𝜁))))

Proof of Theorem exp12bd

Step	Hyp	Ref	Expression
1		exp12bd.1	. 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜃) ↔ ((𝜏 ∧ 𝜂) → 𝜁)))
2		impexp 451	. 2 ⊢ (((𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜓 → (𝜒 → 𝜃)))
3		impexp 451	. 2 ⊢ (((𝜏 ∧ 𝜂) → 𝜁) ↔ (𝜏 → (𝜂 → 𝜁)))
4	1, 2, 3	3bitr3g 313	1 ⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) ↔ (𝜏 → (𝜂 → 𝜁))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397

This theorem is referenced by: (None)