Theorem ss2rexv 4055

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 4053	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦 ∈ 𝐴 𝜑 → ∃𝑦 ∈ 𝐵 𝜑))
2	1	reximdv 3170	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
3		ssrexv 4053	. 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 3070   ⊆ wss 3951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-rex 3071  df-ss 3968

This theorem is referenced by:  prpair  47488  grtriprop  47908