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Metamath Proof Explorer

Theorem xorneg1

Description: The connector \/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 27-Jun-2020)

		Ref	Expression
	Assertion	xorneg1	$⊢ (\neg φ ⊻ ψ) \leftrightarrow \neg (φ ⊻ ψ)$

Proof

Step	Hyp	Ref	Expression
1		xorcom	$⊢ (\neg φ ⊻ ψ) \leftrightarrow (ψ ⊻ \neg φ)$
2		xorneg2	$⊢ (ψ ⊻ \neg φ) \leftrightarrow \neg (ψ ⊻ φ)$
3		xorcom	$⊢ (ψ ⊻ φ) \leftrightarrow (φ ⊻ ψ)$
4	2 3	xchbinx	$⊢ (ψ ⊻ \neg φ) \leftrightarrow \neg (φ ⊻ ψ)$
5	1 4	bitri	$⊢ (\neg φ ⊻ ψ) \leftrightarrow \neg (φ ⊻ ψ)$