Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralrot3 Structured version   Visualization version   GIF version

Theorem ralrot3 3353
 Description: Rotate three restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
Assertion
Ref Expression
ralrot3 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝜑)
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑥,𝑦,𝐶   𝑥,𝑧,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑧)

Proof of Theorem ralrot3
StepHypRef Expression
1 ralcom 3346 . . 3 (∀𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑦𝐵 𝜑)
21ralbii 3160 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝐴𝑧𝐶𝑦𝐵 𝜑)
3 ralcom 3346 . 2 (∀𝑥𝐴𝑧𝐶𝑦𝐵 𝜑 ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝜑)
42, 3bitri 278 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wral 3133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-11 2162 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3138 This theorem is referenced by:  rmodislmodlem  19704  rmodislmod  19705  isclmp  23708  ntrneikb  40717  ntrneixb  40718
 Copyright terms: Public domain W3C validator