Metamath Proof Explorer


Theorem syl212anc

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 )
syl212anc.6
|- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et ) ) -> ze )
Assertion syl212anc
|- ( 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 syl212anc.6
 |-  ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et ) ) -> ze )
7 4 5 jca
 |-  ( ph -> ( ta /\ et ) )
8 1 2 3 7 6 syl211anc
 |-  ( ph -> ze )