Theorem orbi1d

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

	Ref	Expression
Hypothesis	bid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	orbi1d	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ) ↔ ( 𝜒 ∨ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	orbi2d	⊢ ( 𝜑 → ( ( 𝜃 ∨ 𝜓 ) ↔ ( 𝜃 ∨ 𝜒 ) ) )
3		orcom	⊢ ( ( 𝜓 ∨ 𝜃 ) ↔ ( 𝜃 ∨ 𝜓 ) )
4		orcom	⊢ ( ( 𝜒 ∨ 𝜃 ) ↔ ( 𝜃 ∨ 𝜒 ) )
5	2 3 4	3bitr4g	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ) ↔ ( 𝜒 ∨ 𝜃 ) ) )