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

Metamath Proof Explorer

Theorem eqrelrdv2

Description: A version of eqrelrdv . (Contributed by Rodolfo Medina, 10-Oct-2010)

		Ref	Expression
	Hypothesis	eqrelrdv2.1	$⊢ φ \to (⟨x, y⟩ \in A \leftrightarrow ⟨x, y⟩ \in B)$
	Assertion	eqrelrdv2	$⊢ ((Rel A \land Rel B) \land φ) \to A = B$

Step	Hyp	Ref	Expression
1		eqrelrdv2.1	$⊢ φ \to (⟨x, y⟩ \in A \leftrightarrow ⟨x, y⟩ \in B)$
2	1	alrimivv	$⊢ φ \to \forall x \forall y (⟨x, y⟩ \in A \leftrightarrow ⟨x, y⟩ \in B)$
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	imbitrrid	$⊢ (Rel A \land Rel B) \to (φ \to A = B)$
5	4	imp	$⊢ ((Rel A \land Rel B) \land φ) \to A = B$