Metamath Proof Explorer


Theorem syl333anc

Description: A syllogism inference combined with contraction. (Contributed by NM, 10-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 )
syl233anc.8
|- ( ph -> rh )
syl333anc.9
|- ( ph -> mu )
syl333anc.10
|- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) /\ ( si /\ rh /\ mu ) ) -> la )
Assertion syl333anc
|- ( ph -> la )

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 syl233anc.8
 |-  ( ph -> rh )
9 syl333anc.9
 |-  ( ph -> mu )
10 syl333anc.10
 |-  ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) /\ ( si /\ rh /\ mu ) ) -> la )
11 7 8 9 3jca
 |-  ( ph -> ( si /\ rh /\ mu ) )
12 1 2 3 4 5 6 11 10 syl331anc
 |-  ( ph -> la )