MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralrimia Structured version   Visualization version   GIF version

Theorem ralrimia 3258
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 412 . 2 (𝜑 → (𝑥𝐴𝜓))
41, 3ralrimi 3257 1 (𝜑 → ∀𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1783  wcel 2108  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-ral 3062
This theorem is referenced by:  fompt  7138  ss2iundf  43672  ismnushort  44320  modelaxreplem3  44997  ssrabdf  45120  iunssdf  45161  rnmptssd  45201  dmmptdff  45228  axccd  45234  dmmptdf2  45238  mpteq12daOLD  45249  rnmptbd2lem  45255  rnmptssdf  45261  rnmptbdlem  45262  ralrnmpt3  45266  rnmptssbi  45267  fconst7  45271  fmptdff  45278  rnmptssdff  45282  infleinf2  45425  unb2ltle  45426  uzublem  45441  cvgcaule  45502  climinf3  45731  limsupequzlem  45737  limsupre3uzlem  45750  climisp  45761  climrescn  45763  climxrrelem  45764  climxrre  45765  climxlim2lem  45860  dvnprodlem1  45961  saliunclf  46337  meaiuninc3v  46499  fsupdm  46857  finfdm  46861
  Copyright terms: Public domain W3C validator