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Metamath Proof Explorer

Theorem ralimdv

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90 ( alim ). (Contributed by NM, 8-Oct-2003)

		Ref	Expression
	Hypothesis	ralimdv.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	Assertion	ralimdv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralimdv.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2	1	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
3	2	ralimdva	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )