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Theorem r19.26m 2461
Description: Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (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 1386 . 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 2328 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
3 df-ral 2328 . . 3  |-  ( A. x  e.  B  ps  <->  A. x ( x  e.  B  ->  ps )
)
42, 3anbi12i 441 . 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 180 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 set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    e. wcel 1409   A.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: (None)
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