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Theorem 3anim123d 1251
Description: Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
3anim123d.1  |-  ( ph  ->  ( ps  ->  ch ) )
3anim123d.2  |-  ( ph  ->  ( th  ->  ta ) )
3anim123d.3  |-  ( ph  ->  ( et  ->  ze )
)
Assertion
Ref Expression
3anim123d  |-  ( ph  ->  ( ( ps  /\  th 
/\  et )  -> 
( ch  /\  ta  /\ 
ze ) ) )

Proof of Theorem 3anim123d
StepHypRef Expression
1 3anim123d.1 . . . 4  |-  ( ph  ->  ( ps  ->  ch ) )
2 3anim123d.2 . . . 4  |-  ( ph  ->  ( th  ->  ta ) )
31, 2anim12d 328 . . 3  |-  ( ph  ->  ( ( ps  /\  th )  ->  ( ch  /\ 
ta ) ) )
4 3anim123d.3 . . 3  |-  ( ph  ->  ( et  ->  ze )
)
53, 4anim12d 328 . 2  |-  ( ph  ->  ( ( ( ps 
/\  th )  /\  et )  ->  ( ( ch 
/\  ta )  /\  ze ) ) )
6 df-3an 922 . 2  |-  ( ( ps  /\  th  /\  et )  <->  ( ( ps 
/\  th )  /\  et ) )
7 df-3an 922 . 2  |-  ( ( ch  /\  ta  /\  ze )  <->  ( ( ch 
/\  ta )  /\  ze ) )
85, 6, 73imtr4g 203 1  |-  ( ph  ->  ( ( ps  /\  th 
/\  et )  -> 
( ch  /\  ta  /\ 
ze ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ 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:  hb3and  1420  pofun  4075  soss  4077  wessep  4328  isopolem  5492  isosolem  5494  issmo2  5938  smores  5941
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