Theorem ralimdva

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90. (Contributed by NM, 22-May-1999) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralimdva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
2	1	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
3	2	a2d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) → ( 𝑥 ∈ 𝐴 → 𝜒 ) ) )
4	3	ralimdv2	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )