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Theorem ralrot3 3358
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 3351 . . 3 (∀𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑦𝐵 𝜑)
21ralbii 3162 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝐴𝑧𝐶𝑦𝐵 𝜑)
3 ralcom 3351 . 2 (∀𝑥𝐴𝑧𝐶𝑦𝐵 𝜑 ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝜑)
42, 3bitri 276 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wral 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-11 2151
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-ral 3140
This theorem is referenced by:  rmodislmodlem  19630  rmodislmod  19631  isclmp  23628  ntrneikb  40322  ntrneixb  40323
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