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Theorem ralrimia 3240
Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
ralrimia.1 𝑥𝜑
ralrimia.2 ((𝜑𝑥𝐴) → 𝜓)
Assertion
Ref Expression
ralrimia (𝜑 → ∀𝑥𝐴 𝜓)

Proof of Theorem ralrimia
StepHypRef Expression
1 ralrimia.1 . 2 𝑥𝜑
2 ralrimia.2 . . 3 ((𝜑𝑥𝐴) → 𝜓)
32ex 414 . 2 (𝜑 → (𝑥𝐴𝜓))
41, 3ralrimi 3239 1 (𝜑 → ∀𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wnf 1786  wcel 2107  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787  df-ral 3062
This theorem is referenced by:  ss2iundf  42019  ismnushort  42669  ssrabdf  43413  iunssdf  43459  rnmptssd  43504  dmmptdff  43531  axccd  43537  dmmptdf2  43545  mpteq12daOLD  43556  rnmptbd2lem  43563  rnmptssdf  43569  rnmptbdlem  43570  ralrnmpt3  43574  rnmptssbi  43576  fconst7  43580  fmptdff  43587  rnmptssdff  43591  infleinf2  43735  unb2ltle  43736  uzublem  43751  cvgcaule  43813  climinf3  44043  limsupequzlem  44049  limsupre3uzlem  44062  climisp  44073  climrescn  44075  climxrrelem  44076  climxrre  44077  climxlim2lem  44172  saliunclf  44649  meaiuninc3v  44811  fsupdm  45169  finfdm  45173
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