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Theorem r2al 3193
Description: Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 9-Jan-2020.)
Assertion
Ref Expression
r2al (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem r2al
StepHypRef Expression
1 19.21v 1937 . 2 (∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
21r2allem 3140 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  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
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-ral 3060
This theorem is referenced by:  r2ex  3194  r3al  3195  ralcom  3287  nfra2w  3297  moel  3400  raliunxp  5853  codir  6143  qfto  6144  dfpo2  6318  fununi  6643  dff13  7275  mpo2eqb  7565  frpoins3xpg  8164  xpord2indlem  8171  tz7.48lem  8480  qliftfun  8841  zorn2lem4  10537  isirred2  20438  isdomn3  20732  cnmpt12  23691  cnmpt22  23698  dchrelbas3  27297  cvmlift2lem12  35299  dfso2  35735  r2alan  38231  inxpss  38293  inxpss3  38296  iscnrm3lem2  48731  joindm2  48765  meetdm2  48767
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