Metamath Proof Explorer


Theorem syl311anc

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 )
syl311anc.6
|- ( ( ( ps /\ ch /\ th ) /\ ta /\ et ) -> ze )
Assertion syl311anc
|- ( ph -> ze )

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 syl311anc.6
 |-  ( ( ( ps /\ ch /\ th ) /\ ta /\ et ) -> ze )
7 1 2 3 3jca
 |-  ( ph -> ( ps /\ ch /\ th ) )
8 7 4 5 6 syl3anc
 |-  ( ph -> ze )