Theorem ss2rexv 4067

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ss2rexv

Step	Hyp	Ref	Expression
1		ssrexv 4065	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦 ∈ 𝐴 𝜑 → ∃𝑦 ∈ 𝐵 𝜑))
2	1	reximdv 3168	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
3		ssrexv 4065	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝜑))
4	2, 3	syld 47	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4  ∃wrex 3068   ⊆ wss 3963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-rex 3069  df-ss 3980

This theorem is referenced by:  prpair  47426  grtriprop  47846