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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 → ( 𝑥 ∈ 𝐴 → 𝜓 ) )
Assertion	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ralrimi.1	⊢ Ⅎ 𝑥 𝜑
2		ralrimi.2	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) )
3	1	nf5ri	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
4	3 2	hbralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )