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Theorem ss2ralv 3962
 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 3960 . . 3 (𝐴𝐵 → (∀𝑦𝐵 𝜑 → ∀𝑦𝐴 𝜑))
21ralimdv 3147 . 2 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐵 𝜑 → ∀𝑥𝐵𝑦𝐴 𝜑))
3 ssralv 3960 . 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 3107   ⊆ wss 3865 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-ral 3112  df-in 3872  df-ss 3880 This theorem is referenced by:  poss  5371  soss  5388  dffi3  8748  isercolllem1  14859  cfilres  23586  lgsdchr  25617  dffltz  38789
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