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Theorem ralimdaa 3258
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 3255 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 ralim 3084 . 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 1780  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-ral 3060
This theorem is referenced by:  ralbida  3268  eltsk2g  10789  ptcnplem  23645  poimirlem26  37633  allbutfifvre  45631  climleltrp  45632  fnlimabslt  45635  limsupub2  45768  liminflbuz2  45771  xlimmnfvlem1  45788  xlimmnfvlem2  45789  xlimpnfvlem1  45792  xlimpnfvlem2  45793  stoweidlem61  46017  stoweid  46019  fourierdlem73  46135  smflimlem2  46728
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