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Theorem ralimdaa 3211
 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 416 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 3210 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 ralim 3157 . 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 399  Ⅎwnf 1785   ∈ wcel 2115  ∀wral 3133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2179 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-ral 3138 This theorem is referenced by:  eltsk2g  10171  ptcnplem  22233  poimirlem26  35032  allbutfifvre  42248  climleltrp  42249  fnlimabslt  42252  limsupub2  42385  liminflbuz2  42388  stoweidlem61  42634  stoweid  42636  fourierdlem73  42752  smflimlem2  43336
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