Theorem sylnib 687

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 683	1 |- ( ph -> -. ch )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117

This theorem is used by:  sylnibr  688  neqcomd  2243  inssdif0imOLD  3593  undifexmid  4332  ordtriexmidlem2  4669  dmsn0el  5260  fidifsnen  7178  ctssdccl  7457  nninfwlpoimlemginf  7522  onntri35  7602  onntri45  7606  2omotaplemap  7629  exmidapne  7632  ltpopr  7968  caucvgprprlemnbj  8066  xrlttri3  10215  fzneuz  10524  iseqf1olemqcl  10956  iseqf1olemnab  10958  iseqf1olemab  10959  exp3val  10998  ballotfilemimin  13306  ballotfilemfrcn0  13330  psrlidmfi  15208  bposlem9  16336  pwle2  17250  wexmiddiffilem  17265  wexmiddifxylem  17267