Theorem sylnib 677

Description: A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)

Hypotheses

Ref	Expression
sylnib.1	|- ( ph -> -. ps )
sylnib.2	|- ( ps <-> ch )

Assertion

Ref	Expression
sylnib	|- ( ph -> -. ch )

Proof of Theorem sylnib

Step	Hyp	Ref	Expression
1		sylnib.1	. 2 |- ( ph -> -. ps )
2		sylnib.2	. . 3 |- ( ps <-> ch )
3	2	a1i 9	. 2 |- ( ph -> ( ps <-> ch ) )
4	1, 3	mtbid 673	1 |- ( ph -> -. ch )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sylnibr  678  neqcomd  2198  inssdif0im  3515  undifexmid  4223  ordtriexmidlem2  4553  dmsn0el  5136  fidifsnen  6928  ctssdccl  7172  nninfwlpoimlemginf  7237  onntri35  7299  onntri45  7303  2omotaplemap  7319  exmidapne  7322  ltpopr  7657  caucvgprprlemnbj  7755  xrlttri3  9866  fzneuz  10170  iseqf1olemqcl  10573  iseqf1olemnab  10575  iseqf1olemab  10576  exp3val  10615  pwle2  15559