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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	$⊢ φ \leftrightarrow ψ$
	Assertion	orbi2i	$⊢ (χ \lor φ) \leftrightarrow (χ \lor ψ)$

Proof

Step	Hyp	Ref	Expression
1		orbi2i.1	$⊢ φ \leftrightarrow ψ$
2	1	biimpi	$⊢ φ \to ψ$
3	2	orim2i	$⊢ (χ \lor φ) \to (χ \lor ψ)$
4	1	biimpri	$⊢ ψ \to φ$
5	4	orim2i	$⊢ (χ \lor ψ) \to (χ \lor φ)$
6	3 5	impbii	$⊢ (χ \lor φ) \leftrightarrow (χ \lor ψ)$