This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem eqrelriv

Description: Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012)

		Ref	Expression
	Hypothesis	eqrelriv.1	$⊢ ⟨x, y⟩ \in A \leftrightarrow ⟨x, y⟩ \in B$
	Assertion	eqrelriv	$⊢ (Rel A \land Rel B) \to A = B$

Step	Hyp	Ref	Expression
1		eqrelriv.1	$⊢ ⟨x, y⟩ \in A \leftrightarrow ⟨x, y⟩ \in B$
2	1	gen2	$⊢ \forall x \forall y (⟨x, y⟩ \in A \leftrightarrow ⟨x, y⟩ \in B)$
3		eqrel	$⊢ (Rel A \land Rel B) \to (A = B \leftrightarrow \forall x \forall y (⟨x, y⟩ \in A \leftrightarrow ⟨x, y⟩ \in B))$
4	2 3	mpbiri	$⊢ (Rel A \land Rel B) \to A = B$