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Metamath Proof Explorer

Theorem ralrimi

Description: Inference from Theorem 19.21 of Margaris p. 90 (restricted quantifier version). For a version based on fewer axioms see ralrimiv . (Contributed by NM, 10-Oct-1999) Shortened after introduction of hbralrimi . (Revised by Wolf Lammen, 4-Dec-2019)

	Ref	Expression
Hypotheses	ralrimi.1	$⊢ Ⅎ x φ$
	ralrimi.2	$⊢ φ \to (x \in A \to ψ)$
Assertion	ralrimi	$⊢ φ \to \forall x \in A ψ$

Proof

Step	Hyp	Ref	Expression
1		ralrimi.1	$⊢ Ⅎ x φ$
2		ralrimi.2	$⊢ φ \to (x \in A \to ψ)$
3	1	nf5ri	$⊢ φ \to \forall x φ$
4	3 2	hbralrimi	$⊢ φ \to \forall x \in A ψ$