Theorem syl23anc 1235

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 )
syl23anc.6	|- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) ) -> ze )

Assertion

Ref	Expression
syl23anc	|- ( ph -> ze )

Proof of Theorem syl23anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. . 3 |- ( ph -> ps )
2		sylXanc.2	. . 3 |- ( ph -> ch )
3	1, 2	jca 304	. 2 |- ( ph -> ( ps /\ ch ) )
4		sylXanc.3	. 2 |- ( ph -> th )
5		sylXanc.4	. 2 |- ( ph -> ta )
6		sylXanc.5	. 2 |- ( ph -> et )
7		syl23anc.6	. 2 |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) ) -> ze )
8	3, 4, 5, 6, 7	syl13anc 1230	1 |- ( ph -> ze )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 970

This theorem is referenced by:  div2subapd  8734  gcdaddm  11917  restopn2  12823