Theorem bnj1541

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1541.1	$$⊢\phi \leftrightarrow \left(\psi \wedge A\ne B\right)$$
	bnj1541.2	$$⊢\neg \phi$$
Assertion	bnj1541	$$⊢\psi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		bnj1541.1	$$⊢\phi \leftrightarrow \left(\psi \wedge A\ne B\right)$$
2		bnj1541.2	$$⊢\neg \phi$$
3	2 1	mtbi	$$⊢\neg \left(\psi \wedge A\ne B\right)$$
4	3	imnani	$$⊢\psi \to \neg A\ne B$$
5		nne	$$⊢\neg A\ne B\leftrightarrow A=B$$
6	4 5	sylib	$$⊢\psi \to A=B$$