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Metamath Proof Explorer

Theorem ralanid

Description: Cancellation law for restricted universal quantification. (Contributed by Peter Mazsa, 30-Dec-2018) (Proof shortened by Wolf Lammen, 29-Jun-2023)

		Ref	Expression
	Assertion	ralanid	$⊢ \forall x \in A (x \in A \land φ) \leftrightarrow \forall x \in A φ$

Proof

Step	Hyp	Ref	Expression
1		ibar	$⊢ x \in A \to (φ \leftrightarrow (x \in A \land φ))$
2	1	bicomd	$⊢ x \in A \to ((x \in A \land φ) \leftrightarrow φ)$
3	2	ralbiia	$⊢ \forall x \in A (x \in A \land φ) \leftrightarrow \forall x \in A φ$