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Metamath Proof Explorer

Theorem notbid

Description: Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994)

		Ref	Expression
	Hypothesis	notbid.1	$⊢ φ \to (ψ \leftrightarrow χ)$
	Assertion	notbid	$⊢ φ \to (\neg ψ \leftrightarrow \neg χ)$

Step	Hyp	Ref	Expression
1		notbid.1	$⊢ φ \to (ψ \leftrightarrow χ)$
2		notnotb	$⊢ ψ \leftrightarrow \neg \neg ψ$
3		notnotb	$⊢ χ \leftrightarrow \neg \neg χ$
4	1 2 3	3bitr3g	$⊢ φ \to (\neg \neg ψ \leftrightarrow \neg \neg χ)$
5	4	con4bid	$⊢ φ \to (\neg ψ \leftrightarrow \neg χ)$