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Theorem ralimdaa 2941
 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 449 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 2940 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 ralim 2932 . 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 383  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-ral 2901 This theorem is referenced by:  eltsk2g  9452  ptcnplem  21234  poimirlem26  32605  allbutfifvre  38742  climleltrp  38743  fnlimabslt  38746  stoweidlem61  38954  stoweid  38956  fourierdlem73  39072  smflimlem2  39658
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