Metamath Proof Explorer


Theorem syl133anc

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012)

Ref Expression
Hypotheses syl3anc.1
|- ( ph -> ps )
syl3anc.2
|- ( ph -> ch )
syl3anc.3
|- ( ph -> th )
syl3Xanc.4
|- ( ph -> ta )
syl23anc.5
|- ( ph -> et )
syl33anc.6
|- ( ph -> ze )
syl133anc.7
|- ( ph -> si )
syl133anc.8
|- ( ( ps /\ ( ch /\ th /\ ta ) /\ ( et /\ ze /\ si ) ) -> rh )
Assertion syl133anc
|- ( ph -> rh )

Proof

Step Hyp Ref Expression
1 syl3anc.1
 |-  ( ph -> ps )
2 syl3anc.2
 |-  ( ph -> ch )
3 syl3anc.3
 |-  ( ph -> th )
4 syl3Xanc.4
 |-  ( ph -> ta )
5 syl23anc.5
 |-  ( ph -> et )
6 syl33anc.6
 |-  ( ph -> ze )
7 syl133anc.7
 |-  ( ph -> si )
8 syl133anc.8
 |-  ( ( ps /\ ( ch /\ th /\ ta ) /\ ( et /\ ze /\ si ) ) -> rh )
9 5 6 7 3jca
 |-  ( ph -> ( et /\ ze /\ si ) )
10 1 2 3 4 9 8 syl131anc
 |-  ( ph -> rh )