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Metamath Proof Explorer

Theorem necon4abid

Description: Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008) (Proof shortened by Wolf Lammen, 24-Nov-2019)

		Ref	Expression
	Hypothesis	necon4abid.1	$⊢ φ \to (A \neq B \leftrightarrow \neg ψ)$
	Assertion	necon4abid	$⊢ φ \to (A = B \leftrightarrow ψ)$

Step	Hyp	Ref	Expression
1		necon4abid.1	$⊢ φ \to (A \neq B \leftrightarrow \neg ψ)$
2		notnotb	$⊢ ψ \leftrightarrow \neg \neg ψ$
3	1	necon1bbid	$⊢ φ \to (\neg \neg ψ \leftrightarrow A = B)$
4	2 3	bitr2id	$⊢ φ \to (A = B \leftrightarrow ψ)$