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Theorem 3anbi123i 1128
Description: Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
Hypotheses
Ref Expression
bi3.1  |-  ( ph  <->  ps )
bi3.2  |-  ( ch  <->  th )
bi3.3  |-  ( ta  <->  et )
Assertion
Ref Expression
3anbi123i  |-  ( (
ph  /\  ch  /\  ta ) 
<->  ( ps  /\  th  /\  et ) )

Proof of Theorem 3anbi123i
StepHypRef Expression
1 bi3.1 . . . 4  |-  ( ph  <->  ps )
2 bi3.2 . . . 4  |-  ( ch  <->  th )
31, 2anbi12i 448 . . 3  |-  ( (
ph  /\  ch )  <->  ( ps  /\  th )
)
4 bi3.3 . . 3  |-  ( ta  <->  et )
53, 4anbi12i 448 . 2  |-  ( ( ( ph  /\  ch )  /\  ta )  <->  ( ( ps  /\  th )  /\  et ) )
6 df-3an 922 . 2  |-  ( (
ph  /\  ch  /\  ta ) 
<->  ( ( ph  /\  ch )  /\  ta )
)
7 df-3an 922 . 2  |-  ( ( ps  /\  th  /\  et )  <->  ( ( ps 
/\  th )  /\  et ) )
85, 6, 73bitr4i 210 1  |-  ( (
ph  /\  ch  /\  ta ) 
<->  ( ps  /\  th  /\  et ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    /\ w3a 920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-3an 922
This theorem is referenced by:  3anbi1i  1130  3anbi2i  1131  3anbi3i  1132  syl3anb  1213  ne3anior  2337
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