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Theorem bnj887 28795
Description:  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj887.1  |-  ( ph  <->  ph' )
bnj887.2  |-  ( ps  <->  ps' )
bnj887.3  |-  ( ch  <->  ch' )
bnj887.4  |-  ( th  <->  th' )
Assertion
Ref Expression
bnj887  |-  ( (
ph  /\  ps  /\  ch  /\ 
th )  <->  ( ph'  /\  ps'  /\  ch'  /\  th' ) )

Proof of Theorem bnj887
StepHypRef Expression
1 bnj887.1 . . . 4  |-  ( ph  <->  ph' )
2 bnj887.2 . . . 4  |-  ( ps  <->  ps' )
3 bnj887.3 . . . 4  |-  ( ch  <->  ch' )
41, 2, 33anbi123i 1140 . . 3  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ph'  /\  ps'  /\  ch' ) )
5 bnj887.4 . . 3  |-  ( th  <->  th' )
64, 5anbi12i 678 . 2  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  th ) 
<->  ( ( ph'  /\  ps'  /\  ch' )  /\  th' ) )
7 df-bnj17 28712 . 2  |-  ( (
ph  /\  ps  /\  ch  /\ 
th )  <->  ( ( ph  /\  ps  /\  ch )  /\  th ) )
8 df-bnj17 28712 . 2  |-  ( ( ph'  /\  ps'  /\  ch'  /\  th' )  <->  ( ( ph' 
/\  ps'  /\  ch' )  /\  th' ) )
96, 7, 83bitr4i 268 1  |-  ( (
ph  /\  ps  /\  ch  /\ 
th )  <->  ( ph'  /\  ps'  /\  ch'  /\  th' ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    /\ w-bnj17 28711
This theorem is referenced by:  bnj1040  29002  bnj1128  29020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-bnj17 28712
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