Theorem ss2rexv 4003

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 4001	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦 ∈ 𝐴 𝜑 → ∃𝑦 ∈ 𝐵 𝜑))
2	1	reximdv 3178	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
3		ssrexv 4001	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝜑))
4	2, 3	syld 48	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝜑))

This proof depends on syntax axioms:   → wi 4  ∃wrex 3087   ⊆ wss 3899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-rex 3088  df-ss 3916

This theorem is used by:  prpair  48527  grtriprop  48983