Theorem syl32anc 1177

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

Hypotheses

Ref	Expression
sylXanc.1	|- ( ph -> ps )
sylXanc.2	|- ( ph -> ch )
sylXanc.3	|- ( ph -> th )
sylXanc.4	|- ( ph -> ta )
sylXanc.5	|- ( ph -> et )
syl32anc.6	|- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) ) -> ze )

Assertion

Ref	Expression
syl32anc	|- ( ph -> ze )

Proof of Theorem syl32anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 |- ( ph -> ps )
2		sylXanc.2	. 2 |- ( ph -> ch )
3		sylXanc.3	. 2 |- ( ph -> th )
4		sylXanc.4	. . 3 |- ( ph -> ta )
5		sylXanc.5	. . 3 |- ( ph -> et )
6	4, 5	jca 300	. 2 |- ( ph -> ( ta /\ et ) )
7		syl32anc.6	. 2 |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) ) -> ze )
8	1, 2, 3, 6, 7	syl31anc 1172	1 |- ( ph -> ze )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-3an 921

This theorem is referenced by:  ioom  9269  modifeq2int  9388  modaddmodup  9389  exple1  9532  leexp2rd  9635  facubnd  9672  permnn  9698  dfabsmax  10103  dvdsadd2b  10242  dvdsmulgcd  10414  sqgcd  10418  bezoutr  10421  cncongr2  10486  pw2dvds  10544