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Theorem ss2ralv 4013
Description: Two quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023.)
Assertion
Ref Expression
ss2ralv (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵 𝜑 → ∀𝑥𝐴𝑦𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ss2ralv
StepHypRef Expression
1 ssralv 4011 . . 3 (𝐴𝐵 → (∀𝑦𝐵 𝜑 → ∀𝑦𝐴 𝜑))
21ralimdv 3163 . 2 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵 𝜑 → ∀𝑥𝐵𝑦𝐴 𝜑))
3 ssralv 4011 . 2 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐴 𝜑 → ∀𝑥𝐴𝑦𝐴 𝜑))
42, 3syld 47 1 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵 𝜑 → ∀𝑥𝐴𝑦𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wral 3061  wss 3911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3446  df-in 3918  df-ss 3928
This theorem is referenced by:  poss  5548  soss  5566  dffi3  9372  isercolllem1  15555  cfilres  24676  lgsdchr  26719  dffltz  41015
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