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Theorem ralrimia 3245
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 411 . 2 (𝜑 → (𝑥𝐴𝜓))
41, 3ralrimi 3244 1 (𝜑 → ∀𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wnf 1777  wcel 2098  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-nf 1778  df-ral 3051
This theorem is referenced by:  fompt  7131  ss2iundf  43263  ismnushort  43912  ssrabdf  44653  iunssdf  44698  rnmptssd  44740  dmmptdff  44767  axccd  44773  dmmptdf2  44777  mpteq12daOLD  44788  rnmptbd2lem  44794  rnmptssdf  44800  rnmptbdlem  44801  ralrnmpt3  44805  rnmptssbi  44807  fconst7  44811  fmptdff  44818  rnmptssdff  44822  infleinf2  44966  unb2ltle  44967  uzublem  44982  cvgcaule  45044  climinf3  45274  limsupequzlem  45280  limsupre3uzlem  45293  climisp  45304  climrescn  45306  climxrrelem  45307  climxrre  45308  climxlim2lem  45403  saliunclf  45880  meaiuninc3v  46042  fsupdm  46400  finfdm  46404
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