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Theorem 19.26-3an 1585
Description: Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
19.26-3an  |-  ( A. x ( ph  /\  ps  /\  ch )  <->  ( A. x ph  /\  A. x ps  /\  A. x ch ) )

Proof of Theorem 19.26-3an
StepHypRef Expression
1 19.26 1583 . . 3  |-  ( A. x ( ( ph  /\ 
ps )  /\  ch ) 
<->  ( A. x (
ph  /\  ps )  /\  A. x ch )
)
2 19.26 1583 . . . 4  |-  ( A. x ( ph  /\  ps )  <->  ( A. x ph  /\  A. x ps ) )
32anbi1i 676 . . 3  |-  ( ( A. x ( ph  /\ 
ps )  /\  A. x ch )  <->  ( ( A. x ph  /\  A. x ps )  /\  A. x ch ) )
41, 3bitri 240 . 2  |-  ( A. x ( ( ph  /\ 
ps )  /\  ch ) 
<->  ( ( A. x ph  /\  A. x ps )  /\  A. x ch ) )
5 df-3an 936 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
65albii 1556 . 2  |-  ( A. x ( ph  /\  ps  /\  ch )  <->  A. x
( ( ph  /\  ps )  /\  ch )
)
7 df-3an 936 . 2  |-  ( ( A. x ph  /\  A. x ps  /\  A. x ch )  <->  ( ( A. x ph  /\  A. x ps )  /\  A. x ch ) )
84, 6, 73bitr4i 268 1  |-  ( A. x ( ph  /\  ps  /\  ch )  <->  ( A. x ph  /\  A. x ps  /\  A. x ch ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530
This theorem is referenced by:  alrim3con13v  28595  19.21a3con13vVD  28944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936
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