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Metamath Proof Explorer

Theorem syl9r

Description: A nested syllogism inference with different antecedents. (Contributed by NM, 14-May-1993)

Ref Expression
Hypotheses syl9r.1 
|- ( ph -> ( ps -> ch ) )
syl9r.2 
|- ( th -> ( ch -> ta ) )
Assertion syl9r 
|- ( th -> ( ph -> ( ps -> ta ) ) )

Proof

Step Hyp Ref Expression
1 syl9r.1 
 |-  ( ph -> ( ps -> ch ) )
2 syl9r.2 
 |-  ( th -> ( ch -> ta ) )
3 1 2 syl9 
 |-  ( ph -> ( th -> ( ps -> ta ) ) )
4 3 com12 
 |-  ( th -> ( ph -> ( ps -> ta ) ) )