Theorem sylnibr 679

Description: A mixed syllogism inference from an implication and a biconditional. Useful for substituting an consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)

Hypotheses

Ref	Expression
sylnibr.1	|- ( ph -> -. ps )
sylnibr.2	|- ( ch <-> ps )

Assertion

Ref	Expression
sylnibr	|- ( ph -> -. ch )

Proof of Theorem sylnibr

Step	Hyp	Ref	Expression
1		sylnibr.1	. 2 |- ( ph -> -. ps )
2		sylnibr.2	. . 3 |- ( ch <-> ps )
3	2	bicomi 132	. 2 |- ( ps <-> ch )
4	1, 3	sylnib 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 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  rexnalim  2495  nssr  3253  difdif  3298  unssin  3412  inssun  3413  undif3ss  3434  ssdif0im  3525  dcun  3570  prneimg  3815  iundif2ss  3993  nssssr  4266  pofun  4359  frirrg  4397  regexmidlem1  4581  dcdifsnid  6590  unfidisj  7019  fidcenumlemrks  7055  difinfsn  7202  pw1nel3  7343  addnqprlemfl  7672  addnqprlemfu  7673  mulnqprlemfl  7688  mulnqprlemfu  7689  cauappcvgprlemladdru  7769  caucvgprprlemaddq  7821  fzpreddisj  10193  fprodntrivap  11895  pw2dvdslemn  12487  isnsgrp  13238  ivthinclemdisj  15112  dvply1  15237  lgseisenlem1  15547  lgsquadlem3  15556  structiedg0val  15637  pwtrufal  15938  pw1nct  15944  nninfsellemsuc  15953