Theorem sylnib 682

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 678	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 619  ax-in2 620

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sylnibr  683  neqcomd  2236  inssdif0im  3562  undifexmid  4283  ordtriexmidlem2  4618  dmsn0el  5206  fidifsnen  7056  ctssdccl  7309  nninfwlpoimlemginf  7374  onntri35  7454  onntri45  7458  2omotaplemap  7475  exmidapne  7478  ltpopr  7814  caucvgprprlemnbj  7912  xrlttri3  10031  fzneuz  10335  iseqf1olemqcl  10760  iseqf1olemnab  10762  iseqf1olemab  10763  exp3val  10802  pwle2  16599