Theorem orbi12d

Description: Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypotheses	bi12d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	bi12d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
Assertion	orbi12d	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ) ↔ ( 𝜒 ∨ 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bi12d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		bi12d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
3	1	orbi1d	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ) ↔ ( 𝜒 ∨ 𝜃 ) ) )
4	2	orbi2d	⊢ ( 𝜑 → ( ( 𝜒 ∨ 𝜃 ) ↔ ( 𝜒 ∨ 𝜏 ) ) )
5	3 4	bitrd	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ) ↔ ( 𝜒 ∨ 𝜏 ) ) )