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Theorem ralcom3 3099
 Description: A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.)
Assertion
Ref Expression
ralcom3 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))

Proof of Theorem ralcom3
StepHypRef Expression
1 pm2.04 90 . . 3 ((𝑥𝐴 → (𝑥𝐵𝜑)) → (𝑥𝐵 → (𝑥𝐴𝜑)))
21ralimi2 2945 . 2 (∀𝑥𝐴 (𝑥𝐵𝜑) → ∀𝑥𝐵 (𝑥𝐴𝜑))
3 pm2.04 90 . . 3 ((𝑥𝐵 → (𝑥𝐴𝜑)) → (𝑥𝐴 → (𝑥𝐵𝜑)))
43ralimi2 2945 . 2 (∀𝑥𝐵 (𝑥𝐴𝜑) → ∀𝑥𝐴 (𝑥𝐵𝜑))
52, 4impbii 199 1 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1987  ∀wral 2908 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734 This theorem depends on definitions:  df-bi 197  df-ral 2913 This theorem is referenced by:  tgss2  20731  ist1-3  21093  isreg2  21121
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