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Metamath Proof Explorer

Theorem orbi2i

Description: Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993) (Proof shortened by Wolf Lammen, 12-Dec-2012)

		Ref	Expression
	Hypothesis	orbi2i.1	⊢ ( 𝜑 ↔ 𝜓 )
	Assertion	orbi2i	⊢ ( ( 𝜒 ∨ 𝜑 ) ↔ ( 𝜒 ∨ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		orbi2i.1	⊢ ( 𝜑 ↔ 𝜓 )
2	1	biimpi	⊢ ( 𝜑 → 𝜓 )
3	2	orim2i	⊢ ( ( 𝜒 ∨ 𝜑 ) → ( 𝜒 ∨ 𝜓 ) )
4	1	biimpri	⊢ ( 𝜓 → 𝜑 )
5	4	orim2i	⊢ ( ( 𝜒 ∨ 𝜓 ) → ( 𝜒 ∨ 𝜑 ) )
6	3 5	impbii	⊢ ( ( 𝜒 ∨ 𝜑 ) ↔ ( 𝜒 ∨ 𝜓 ) )