Theorem pm5.32dra

Description: Reverse distribution of implication over biconditional (deduction form). (Contributed by Zhi Wang, 6-Sep-2024)

	Ref	Expression
Hypothesis	pm5.32dra.1	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) ) )
Assertion	pm5.32dra	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.32dra.1	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) ) )
2		pm5.32	⊢ ( ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) ↔ ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) ) )
3	1 2	sylibr	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) )
4	3	imp	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) )