Theorem orbi2d

Description: Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypothesis	bid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	orbi2d	⊢ ( 𝜑 → ( ( 𝜃 ∨ 𝜓 ) ↔ ( 𝜃 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	imbi2d	⊢ ( 𝜑 → ( ( ¬ 𝜃 → 𝜓 ) ↔ ( ¬ 𝜃 → 𝜒 ) ) )
3		df-or	⊢ ( ( 𝜃 ∨ 𝜓 ) ↔ ( ¬ 𝜃 → 𝜓 ) )
4		df-or	⊢ ( ( 𝜃 ∨ 𝜒 ) ↔ ( ¬ 𝜃 → 𝜒 ) )
5	2 3 4	3bitr4g	⊢ ( 𝜑 → ( ( 𝜃 ∨ 𝜓 ) ↔ ( 𝜃 ∨ 𝜒 ) ) )