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Theorem ralcom3 2866
Description: A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
Assertion
Ref Expression
ralcom3  |-  ( A. x  e.  A  (
x  e.  B  ->  ph )  <->  A. x  e.  B  ( x  e.  A  ->  ph ) )

Proof of Theorem ralcom3
StepHypRef Expression
1 pm2.04 78 . . 3  |-  ( ( x  e.  A  -> 
( x  e.  B  ->  ph ) )  -> 
( x  e.  B  ->  ( x  e.  A  ->  ph ) ) )
21ralimi2 2771 . 2  |-  ( A. x  e.  A  (
x  e.  B  ->  ph )  ->  A. x  e.  B  ( x  e.  A  ->  ph )
)
3 pm2.04 78 . . 3  |-  ( ( x  e.  B  -> 
( x  e.  A  ->  ph ) )  -> 
( x  e.  A  ->  ( x  e.  B  ->  ph ) ) )
43ralimi2 2771 . 2  |-  ( A. x  e.  B  (
x  e.  A  ->  ph )  ->  A. x  e.  A  ( x  e.  B  ->  ph )
)
52, 4impbii 181 1  |-  ( A. x  e.  A  (
x  e.  B  ->  ph )  <->  A. x  e.  B  ( x  e.  A  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1725   A.wral 2698
This theorem is referenced by:  tgss2  17045  ist1-3  17406  isreg2  17434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566
This theorem depends on definitions:  df-bi 178  df-ral 2703
  Copyright terms: Public domain W3C validator