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Theorem syl2an23an 1231
Description: Deduction related to syl3an 1212 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
Hypotheses
Ref Expression
syl2an23an.1  |-  ( ph  ->  ps )
syl2an23an.2  |-  ( ph  ->  ch )
syl2an23an.3  |-  ( ( th  /\  ph )  ->  ta )
syl2an23an.4  |-  ( ( ps  /\  ch  /\  ta )  ->  et )
Assertion
Ref Expression
syl2an23an  |-  ( ( th  /\  ph )  ->  et )

Proof of Theorem syl2an23an
StepHypRef Expression
1 syl2an23an.3 . . 3  |-  ( ( th  /\  ph )  ->  ta )
2 syl2an23an.1 . . . 4  |-  ( ph  ->  ps )
3 syl2an23an.2 . . . 4  |-  ( ph  ->  ch )
4 syl2an23an.4 . . . . 5  |-  ( ( ps  /\  ch  /\  ta )  ->  et )
543exp 1138 . . . 4  |-  ( ps 
->  ( ch  ->  ( ta  ->  et ) ) )
62, 3, 5sylc 61 . . 3  |-  ( ph  ->  ( ta  ->  et ) )
71, 6syl5 32 . 2  |-  ( ph  ->  ( ( th  /\  ph )  ->  et )
)
87anabsi7 546 1  |-  ( ( th  /\  ph )  ->  et )
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: (None)
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