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Theorem ss2rexv 3922
 Description: Two existential quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023.)
Assertion
Ref Expression
ss2rexv (𝐴𝐵 → (∃𝑥𝐴𝑦𝐴 𝜑 → ∃𝑥𝐵𝑦𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ss2rexv
StepHypRef Expression
1 ssrexv 3920 . . 3 (𝐴𝐵 → (∃𝑦𝐴 𝜑 → ∃𝑦𝐵 𝜑))
21reximdv 3212 . 2 (𝐴𝐵 → (∃𝑥𝐴𝑦𝐴 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑))
3 ssrexv 3920 . 2 (𝐴𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐵𝑦𝐵 𝜑))
42, 3syld 47 1 (𝐴𝐵 → (∃𝑥𝐴𝑦𝐴 𝜑 → ∃𝑥𝐵𝑦𝐵 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wrex 3083   ⊆ wss 3825 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-ral 3087  df-rex 3088  df-in 3832  df-ss 3839 This theorem is referenced by:  prpair  42971
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