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Theorem ss2ralv 4053
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 4051 . . 3 (𝐴𝐵 → (∀𝑦𝐵 𝜑 → ∀𝑦𝐴 𝜑))
21ralimdv 3168 . 2 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵 𝜑 → ∀𝑥𝐵𝑦𝐴 𝜑))
3 ssralv 4051 . 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 3060  wss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909
This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3061  df-ss 3967
This theorem is referenced by:  poss  5593  soss  5611  dffi3  9472  isercolllem1  15702  cfilres  25331  lgsdchr  27400  dffltz  42649
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