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	$$⊢\phi \to \left(A\ne B\leftrightarrow \neg \psi \right)$$
Assertion	necon4abid	$$⊢\phi \to \left(A=B\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		necon4abid.1	$$⊢\phi \to \left(A\ne B\leftrightarrow \neg \psi \right)$$
2		notnotb	$$⊢\psi \leftrightarrow \neg \neg \psi$$
3	1	necon1bbid	$$⊢\phi \to \left(\neg \neg \psi \leftrightarrow A=B\right)$$
4	2 3	syl5rbb	$$⊢\phi \to \left(A=B\leftrightarrow \psi \right)$$