Theorem imbi12d

Description: Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 16-May-1993)

	Ref	Expression
Hypotheses	imbi12d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	imbi12d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
Assertion	imbi12d	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imbi12d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		imbi12d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
3	1	imbi1d	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜃 ) ) )
4	2	imbi2d	⊢ ( 𝜑 → ( ( 𝜒 → 𝜃 ) ↔ ( 𝜒 → 𝜏 ) ) )
5	3 4	bitrd	⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜏 ) ) )