Theorem sylnib 672

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 668	1 |- ( ph -> -. ch )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  sylnibr  673  neqcomd  2176  inssdif0im  3483  undifexmid  4180  ordtriexmidlem2  4505  dmsn0el  5082  fidifsnen  6852  ctssdccl  7092  nninfwlpoimlemginf  7156  onntri35  7218  onntri45  7222  ltpopr  7561  caucvgprprlemnbj  7659  xrlttri3  9758  fzneuz  10061  iseqf1olemqcl  10446  iseqf1olemnab  10448  iseqf1olemab  10449  exp3val  10482  pwle2  14116