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Theorem ralimdaa 3259
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.) (Proof shortened by Wolf Lammen, 29-Dec-2019.)
Hypotheses
Ref Expression
ralimdaa.1 𝑥𝜑
ralimdaa.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralimdaa (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Proof of Theorem ralimdaa
StepHypRef Expression
1 ralimdaa.1 . . 3 𝑥𝜑
2 ralimdaa.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32ex 412 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 3256 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 ralim 3085 . 2 (∀𝑥𝐴 (𝜓𝜒) → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
64, 5syl 17 1 (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1782  wcel 2107  wral 3060
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  ax-6 1966  ax-7 2006  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-ral 3061
This theorem is referenced by:  ralbida  3269  eltsk2g  10792  ptcnplem  23630  poimirlem26  37654  allbutfifvre  45695  climleltrp  45696  fnlimabslt  45699  limsupub2  45832  liminflbuz2  45835  xlimmnfvlem1  45852  xlimmnfvlem2  45853  xlimpnfvlem1  45856  xlimpnfvlem2  45857  stoweidlem61  46081  stoweid  46083  fourierdlem73  46199  smflimlem2  46792
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