Theorem ralimdvva

Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90 ( alim ). (Contributed by AV, 27-Nov-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralimdvva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝜓 → 𝜒 ) )
2	1	anassrs	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝜓 → 𝜒 ) )
3	2	ralimdva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝐵 𝜓 → ∀ 𝑦 ∈ 𝐵 𝜒 ) )
4	3	ralimdva	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ) )