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Theorem ralcom3 2494
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 80 . . 3 ((𝑥𝐴 → (𝑥𝐵𝜑)) → (𝑥𝐵 → (𝑥𝐴𝜑)))
21ralimi2 2398 . 2 (∀𝑥𝐴 (𝑥𝐵𝜑) → ∀𝑥𝐵 (𝑥𝐴𝜑))
3 pm2.04 80 . . 3 ((𝑥𝐵 → (𝑥𝐴𝜑)) → (𝑥𝐴 → (𝑥𝐵𝜑)))
43ralimi2 2398 . 2 (∀𝑥𝐵 (𝑥𝐴𝜑) → ∀𝑥𝐴 (𝑥𝐵𝜑))
52, 4impbii 121 1 (∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥𝐵 (𝑥𝐴𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114  df-ral 2328
This theorem is referenced by:  zfregfr  4326
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