Theorem sylnib 678

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 674	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  679  neqcomd  2212  inssdif0im  3536  undifexmid  4253  ordtriexmidlem2  4586  dmsn0el  5171  fidifsnen  6993  ctssdccl  7239  nninfwlpoimlemginf  7304  onntri35  7383  onntri45  7387  2omotaplemap  7404  exmidapne  7407  ltpopr  7743  caucvgprprlemnbj  7841  xrlttri3  9954  fzneuz  10258  iseqf1olemqcl  10681  iseqf1olemnab  10683  iseqf1olemab  10684  exp3val  10723  pwle2  16137