Theorem imbitrdi 161

Description: A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)

Hypotheses

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

Assertion

Ref	Expression
imbitrdi	|- ( ph -> ( ps -> th ) )

Proof of Theorem imbitrdi

Step	Hyp	Ref	Expression
1		imbitrdi.1	. 2 |- ( ph -> ( ps -> ch ) )
2		imbitrdi.2	. . 3 |- ( ch <-> th )
3	2	biimpi 120	. 2 |- ( ch -> th )
4	1, 3	syl6 33	1 |- ( ph -> ( ps -> th ) )

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

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This proof depends on definitions:  df-bi 117

This theorem is used by:  3imtr3g  204  exp4a  366  con2biddc  892  nfalt  1631  alexim  1698  19.36-1  1725  ax11ev  1881  equs5or  1883  necon2bd  2478  necon2d  2479  necon1bbiddc  2483  necon2abiddc  2486  necon2bbiddc  2487  necon4idc  2489  necon4ddc  2492  necon1bddc  2497  spc2gv  2916  spc3gv  2918  mo2icl  3005  reupick  3517  prneimg  3899  invdisj  4123  trin  4241  exmidsssnc  4342  ordsucss  4653  eqbrrdva  4952  elreldm  5010  elres  5101  xp11m  5229  ssrnres  5233  opelf  5563  dffo4  5859  dftpos3  6538  tfr1onlemaccex  6624  tfrcllemaccex  6637  nnaordex  6806  swoer  6840  map0g  6974  mapsn  6978  nneneq  7164  fnfi  7256  prarloclemlo  7867  genprndl  7894  genprndu  7895  cauappcvgprlemladdrl  8030  caucvgprlemladdrl  8051  caucvgsrlemoffres  8173  caucvgsr  8175  nntopi  8267  letr  8414  reapcotr  8934  apcotr  8943  mulext1  8948  lt2msq  9224  nneoor  9758  xrletr  10226  icoshft  10408  hashf1  11308  swrdccatin2  11522  caucvgre  11768  absext  11850  rexico  12009  summodc  12174  gcdeq0  12778  intopsn  13745  znleval  15096  tgcn  15451  cnptoprest  15482  metequiv2  15739  ppiqeq0  16297  bposlem6  16333  bj-nnsn  16983  bj-inf2vnlem2  17219