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Theorem ss2ralv 4032
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 4030 . . 3 (𝐴𝐵 → (∀𝑦𝐵 𝜑 → ∀𝑦𝐴 𝜑))
21ralimdv 3175 . 2 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵 𝜑 → ∀𝑥𝐵𝑦𝐴 𝜑))
3 ssralv 4030 . 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 3135  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ral 3140  df-in 3940  df-ss 3949
This theorem is referenced by:  poss  5469  soss  5486  dffi3  8883  isercolllem1  15009  cfilres  23826  lgsdchr  25858  dffltz  39149
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