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Theorem r19.26m 2984
Description: Version of 19.26 1685 and r19.26 2981 with restricted quantifiers ranging over different classes. (Contributed by NM, 22-Feb-2004.)
Assertion
Ref Expression
r19.26m  |-  ( A. x ( ( x  e.  A  ->  ph )  /\  ( x  e.  B  ->  ps ) )  <->  ( A. x  e.  A  ph  /\  A. x  e.  B  ps ) )

Proof of Theorem r19.26m
StepHypRef Expression
1 19.26 1685 . 2  |-  ( A. x ( ( x  e.  A  ->  ph )  /\  ( x  e.  B  ->  ps ) )  <->  ( A. x ( x  e.  A  ->  ph )  /\  A. x ( x  e.  B  ->  ps )
) )
2 df-ral 2809 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
3 df-ral 2809 . . 3  |-  ( A. x  e.  B  ps  <->  A. x ( x  e.  B  ->  ps )
)
42, 3anbi12i 695 . 2  |-  ( ( A. x  e.  A  ph 
/\  A. x  e.  B  ps )  <->  ( A. x
( x  e.  A  ->  ph )  /\  A. x ( x  e.  B  ->  ps )
) )
51, 4bitr4i 252 1  |-  ( A. x ( ( x  e.  A  ->  ph )  /\  ( x  e.  B  ->  ps ) )  <->  ( A. x  e.  A  ph  /\  A. x  e.  B  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    e. wcel 1823   A.wral 2804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636
This theorem depends on definitions:  df-bi 185  df-an 369  df-ral 2809
This theorem is referenced by: (None)
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