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Theorem 19.26-3an 1649
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 1647 . . 3  |-  ( A. x ( ( ph  /\ 
ps )  /\  ch ) 
<->  ( A. x (
ph  /\  ps )  /\  A. x ch )
)
2 19.26 1647 . . . 4  |-  ( A. x ( ph  /\  ps )  <->  ( A. x ph  /\  A. x ps ) )
32anbi1i 695 . . 3  |-  ( ( A. x ( ph  /\ 
ps )  /\  A. x ch )  <->  ( ( A. x ph  /\  A. x ps )  /\  A. x ch ) )
41, 3bitri 249 . 2  |-  ( A. x ( ( ph  /\ 
ps )  /\  ch ) 
<->  ( ( A. x ph  /\  A. x ps )  /\  A. x ch ) )
5 df-3an 967 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
65albii 1610 . 2  |-  ( A. x ( ph  /\  ps  /\  ch )  <->  A. x
( ( ph  /\  ps )  /\  ch )
)
7 df-3an 967 . 2  |-  ( ( A. x ph  /\  A. x ps  /\  A. x ch )  <->  ( ( A. x ph  /\  A. x ps )  /\  A. x ch ) )
84, 6, 73bitr4i 277 1  |-  ( A. x ( ph  /\  ps  /\  ch )  <->  ( A. x ph  /\  A. x ps  /\  A. x ch ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967
This theorem is referenced by:  alrim3con13v  31239  19.21a3con13vVD  31588
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