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Metamath Proof Explorer

Theorem necon2bbid

Description: Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007) (Proof shortened by Wolf Lammen, 24-Nov-2019)

		Ref	Expression
	Hypothesis	necon2bbid.1	$⊢ φ \to (ψ \leftrightarrow A \neq B)$
	Assertion	necon2bbid	$⊢ φ \to (A = B \leftrightarrow \neg ψ)$

Step	Hyp	Ref	Expression
1		necon2bbid.1	$⊢ φ \to (ψ \leftrightarrow A \neq B)$
2		notnotb	$⊢ ψ \leftrightarrow \neg \neg ψ$
3	1 2	bitr3di	$⊢ φ \to (A \neq B \leftrightarrow \neg \neg ψ)$
4	3	necon4abid	$⊢ φ \to (A = B \leftrightarrow \neg ψ)$