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Theorem ralcom3 3087
Description: A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)
Assertion
Ref Expression
ralcom3 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))

Proof of Theorem ralcom3
StepHypRef Expression
1 bi2.04 386 . 2 ((𝑥𝐴 → (𝑥𝐵𝜑)) ↔ (𝑥𝐵 → (𝑥𝐴𝜑)))
21ralbii2 3079 1 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2099  wral 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804
This theorem depends on definitions:  df-bi 206  df-ral 3052
This theorem is referenced by:  tgss2  22978  ist1-3  23341  isreg2  23369
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