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Theorem a2and 526
Description: Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
Hypotheses
Ref Expression
a2and.1  |-  ( ph  ->  ( ( ps  /\  rh )  ->  ( ta 
->  th ) ) )
a2and.2  |-  ( ph  ->  ( ( ps  /\  rh )  ->  ch )
)
Assertion
Ref Expression
a2and  |-  ( ph  ->  ( ( ( ps 
/\  ch )  ->  ta )  ->  ( ( ps 
/\  rh )  ->  th ) ) )

Proof of Theorem a2and
StepHypRef Expression
1 a2and.2 . . . . . . 7  |-  ( ph  ->  ( ( ps  /\  rh )  ->  ch )
)
21expd 255 . . . . . 6  |-  ( ph  ->  ( ps  ->  ( rh  ->  ch ) ) )
32imdistand 437 . . . . 5  |-  ( ph  ->  ( ( ps  /\  rh )  ->  ( ps 
/\  ch ) ) )
43imp 123 . . . 4  |-  ( (
ph  /\  ( ps  /\  rh ) )  -> 
( ps  /\  ch ) )
5 a2and.1 . . . . 5  |-  ( ph  ->  ( ( ps  /\  rh )  ->  ( ta 
->  th ) ) )
65imp 123 . . . 4  |-  ( (
ph  /\  ( ps  /\  rh ) )  -> 
( ta  ->  th )
)
74, 6embantd 56 . . 3  |-  ( (
ph  /\  ( ps  /\  rh ) )  -> 
( ( ( ps 
/\  ch )  ->  ta )  ->  th ) )
87ex 114 . 2  |-  ( ph  ->  ( ( ps  /\  rh )  ->  ( ( ( ps  /\  ch )  ->  ta )  ->  th ) ) )
98com23 78 1  |-  ( ph  ->  ( ( ( ps 
/\  ch )  ->  ta )  ->  ( ( ps 
/\  rh )  ->  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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