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Theorem r19.26-3 2690
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 936 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
21ralbii 2580 . 2  |-  ( A. x  e.  A  ( ph  /\  ps  /\  ch ) 
<-> 
A. x  e.  A  ( ( ph  /\  ps )  /\  ch )
)
3 r19.26 2688 . 2  |-  ( A. x  e.  A  (
( ph  /\  ps )  /\  ch )  <->  ( A. x  e.  A  ( ph  /\  ps )  /\  A. x  e.  A  ch ) )
4 r19.26 2688 . . . 4  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
54anbi1i 676 . . 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 936 . . 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 243 . 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 262 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:    <-> wb 176    /\ wa 358    /\ w3a 934   A.wral 2556
This theorem is referenced by:  axeuclid  24663  axcontlem8  24671  svli2  25587  bsstrs  26249  nbssntrs  26250  stoweidlem60  27912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-nf 1535  df-ral 2561
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