Theorem sylnib 676

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 672	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 614  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sylnibr  677  neqcomd  2182  inssdif0im  3492  undifexmid  4195  ordtriexmidlem2  4521  dmsn0el  5100  fidifsnen  6872  ctssdccl  7112  nninfwlpoimlemginf  7176  onntri35  7238  onntri45  7242  2omotaplemap  7258  exmidapne  7261  ltpopr  7596  caucvgprprlemnbj  7694  xrlttri3  9799  fzneuz  10103  iseqf1olemqcl  10488  iseqf1olemnab  10490  iseqf1olemab  10491  exp3val  10524  pwle2  14833