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Theorem r19.26-3 2587
Description: Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)
Assertion
Ref Expression
r19.26-3  |-  ( A. x  e.  A  ( ph  /\  ps  /\  ch ) 
<->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps  /\ 
A. x  e.  A  ch ) )

Proof of Theorem r19.26-3
StepHypRef Expression
1 df-3an 965 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
21ralbii 2463 . 2  |-  ( A. x  e.  A  ( ph  /\  ps  /\  ch ) 
<-> 
A. x  e.  A  ( ( ph  /\  ps )  /\  ch )
)
3 r19.26 2583 . 2  |-  ( A. x  e.  A  (
( ph  /\  ps )  /\  ch )  <->  ( A. x  e.  A  ( ph  /\  ps )  /\  A. x  e.  A  ch ) )
4 r19.26 2583 . . . 4  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
54anbi1i 454 . . 3  |-  ( ( A. x  e.  A  ( ph  /\  ps )  /\  A. x  e.  A  ch )  <->  ( ( A. x  e.  A  ph  /\  A. x  e.  A  ps )  /\  A. x  e.  A  ch ) )
6 df-3an 965 . . 3  |-  ( ( A. x  e.  A  ph 
/\  A. x  e.  A  ps  /\  A. x  e.  A  ch )  <->  ( ( A. x  e.  A  ph 
/\  A. x  e.  A  ps )  /\  A. x  e.  A  ch )
)
75, 6bitr4i 186 . 2  |-  ( ( A. x  e.  A  ( ph  /\  ps )  /\  A. x  e.  A  ch )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps  /\ 
A. x  e.  A  ch ) )
82, 3, 73bitri 205 1  |-  ( A. x  e.  A  ( ph  /\  ps  /\  ch ) 
<->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps  /\ 
A. x  e.  A  ch ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 963   A.wral 2435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-4 1490  ax-17 1506
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-ral 2440
This theorem is referenced by: (None)
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