Theorem imbi12d3

Description: Variant of imbi12d2 . (Contributed by Zhi Wang, 30-Aug-2024)

	Ref	Expression
Hypotheses	imbi12d2.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	imbi12d3.2	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) )
Assertion	imbi12d3	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imbi12d2.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		imbi12d3.2	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) )
3	1	pm4.71da	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜓 ∧ 𝜒 ) ) )
4	3 2	sylbid	⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 ↔ 𝜏 ) ) )
5	1 4	imbi12d2	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜏 ) ) )