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Theorem ralcom3 2776
Description: A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
Assertion
Ref Expression
ralcom3 (x A (x Bφ) ↔ x B (x Aφ))

Proof of Theorem ralcom3
StepHypRef Expression
1 pm2.04 76 . . 3 ((x A → (x Bφ)) → (x B → (x Aφ)))
21ralimi2 2686 . 2 (x A (x Bφ) → x B (x Aφ))
3 pm2.04 76 . . 3 ((x B → (x Aφ)) → (x A → (x Bφ)))
43ralimi2 2686 . 2 (x B (x Aφ) → x A (x Bφ))
52, 4impbii 180 1 (x A (x Bφ) ↔ x B (x Aφ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wcel 1710  wral 2614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557
This theorem depends on definitions:  df-bi 177  df-ral 2619
This theorem is referenced by: (None)
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