Theorem reximdvva

Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by AV, 5-Jan-2022)

	Ref	Expression
Hypothesis	ralimdvva.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝜓 → 𝜒 ) )
Assertion	reximdvva	⊢ ( 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	reximdva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∃ 𝑦 ∈ 𝐵 𝜓 → ∃ 𝑦 ∈ 𝐵 𝜒 ) )
4	3	reximdva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜓 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ) )