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Metamath Proof Explorer

Theorem rmov

Description: An at-most-one quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017)

		Ref	Expression
	Assertion	rmov	$⊢ \exists^{} x \in V φ \leftrightarrow \exists^{} x φ$

Step	Hyp	Ref	Expression
1		df-rmo	$⊢ \exists^{} x \in V φ \leftrightarrow \exists^{} x (x \in V \land φ)$
2		vex	$⊢ x \in V$
3	2	biantrur	$⊢ φ \leftrightarrow (x \in V \land φ)$
4	3	mobii	$⊢ \exists^{} x φ \leftrightarrow \exists^{} x (x \in V \land φ)$
5	1 4	bitr4i	$⊢ \exists^{} x \in V φ \leftrightarrow \exists^{} x φ$