Metamath Proof Explorer


Theorem syl213anc

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 )
syl213anc.7
|- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et /\ ze ) ) -> si )
Assertion syl213anc
|- ( ph -> si )

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 syl213anc.7
 |-  ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et /\ ze ) ) -> si )
8 1 2 jca
 |-  ( ph -> ( ps /\ ch ) )
9 8 3 4 5 6 7 syl113anc
 |-  ( ph -> si )