Theorem sylnib 680

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 676	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 617  ax-in2 618

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sylnibr  681  neqcomd  2234  inssdif0im  3559  undifexmid  4277  ordtriexmidlem2  4612  dmsn0el  5198  fidifsnen  7040  ctssdccl  7289  nninfwlpoimlemginf  7354  onntri35  7433  onntri45  7437  2omotaplemap  7454  exmidapne  7457  ltpopr  7793  caucvgprprlemnbj  7891  xrlttri3  10005  fzneuz  10309  iseqf1olemqcl  10733  iseqf1olemnab  10735  iseqf1olemab  10736  exp3val  10775  pwle2  16423