Theorem sylnib 683

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 679	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  684  neqcomd  2237  inssdif0im  3576  undifexmid  4306  ordtriexmidlem2  4642  dmsn0el  5232  fidifsnen  7125  ctssdccl  7402  nninfwlpoimlemginf  7467  onntri35  7547  onntri45  7551  2omotaplemap  7571  exmidapne  7574  ltpopr  7910  caucvgprprlemnbj  8008  xrlttri3  10130  fzneuz  10435  iseqf1olemqcl  10861  iseqf1olemnab  10863  iseqf1olemab  10864  exp3val  10903  pwle2  16772