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Theorem ralcom13 3351
Description: Swap first and third restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)
Assertion
Ref Expression
ralcom13 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑧,𝐵   𝑥,𝑦,𝐶
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑧)

Proof of Theorem ralcom13
StepHypRef Expression
1 ralcom 3346 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑦𝐵𝑥𝐴𝑧𝐶 𝜑)
2 ralcom 3346 . . 3 (∀𝑥𝐴𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑥𝐴 𝜑)
32ralbii 3160 . 2 (∀𝑦𝐵𝑥𝐴𝑧𝐶 𝜑 ↔ ∀𝑦𝐵𝑧𝐶𝑥𝐴 𝜑)
4 ralcom 3346 . 2 (∀𝑦𝐵𝑧𝐶𝑥𝐴 𝜑 ↔ ∀𝑧𝐶𝑦𝐵𝑥𝐴 𝜑)
51, 3, 43bitri 300 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑧𝐶𝑦𝐵𝑥𝐴 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: (None)
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