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  4276  ordtriexmidlem2  4611  dmsn0el  5197  fidifsnen  7028  ctssdccl  7274  nninfwlpoimlemginf  7339  onntri35  7418  onntri45  7422  2omotaplemap  7439  exmidapne  7442  ltpopr  7778  caucvgprprlemnbj  7876  xrlttri3  9989  fzneuz  10293  iseqf1olemqcl  10716  iseqf1olemnab  10718  iseqf1olemab  10719  exp3val  10758  pwle2  16323