Theorem ss2rexv

Description: Two existential quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023)

	Ref	Expression
Assertion	ss2rexv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ssrexv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑦 ∈ 𝐴 𝜑 → ∃ 𝑦 ∈ 𝐵 𝜑 ) )
2	1	reximdv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ) )
3		ssrexv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 𝜑 ) )
4	2 3	syld	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 𝜑 ) )