MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r2al Structured version   Visualization version   GIF version

Theorem r2al 2923
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 1855 . 2 (∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
21r2allem 2921 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473  wcel 1977  wral 2896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ral 2901
This theorem is referenced by:  r3al  2924  r2ex  3043  soss  4977  raliunxp  5183  codir  5435  qfto  5436  fununi  5878  dff13  6416  mpt22eqb  6667  tz7.48lem  7423  qliftfun  7719  zorn2lem4  9204  isirred2  18524  cnmpt12  21280  cnmpt22  21287  dchrelbas3  24763  cvmlift2lem12  30550  dfso2  30897  dfpo2  30898  isdomn3  36801
  Copyright terms: Public domain W3C validator