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Theorem ralcom3 2596
Description: A commutative law for restricted quantifiers that swaps the domain 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 82 . . 3 ((𝑥𝐴 → (𝑥𝐵𝜑)) → (𝑥𝐵 → (𝑥𝐴𝜑)))
21ralimi2 2490 . 2 (∀𝑥𝐴 (𝑥𝐵𝜑) → ∀𝑥𝐵 (𝑥𝐴𝜑))
3 pm2.04 82 . . 3 ((𝑥𝐵 → (𝑥𝐴𝜑)) → (𝑥𝐴 → (𝑥𝐵𝜑)))
43ralimi2 2490 . 2 (∀𝑥𝐵 (𝑥𝐴𝜑) → ∀𝑥𝐴 (𝑥𝐵𝜑))
52, 4impbii 125 1 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1480  wral 2414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425
This theorem depends on definitions:  df-bi 116  df-ral 2419
This theorem is referenced by:  zfregfr  4483  tgss2  12237
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