Theorem mpbii 148

Description: An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)

Hypotheses

Ref	Expression
mpbii.min	|- ps
mpbii.maj	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
mpbii	|- ( ph -> ch )

Proof of Theorem mpbii

Step	Hyp	Ref	Expression
1		mpbii.min	. . 3 |- ps
2	1	a1i 9	. 2 |- ( ph -> ps )
3		mpbii.maj	. 2 |- ( ph -> ( ps <-> ch ) )
4	2, 3	mpbid 147	1 |- ( ph -> ch )

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:  pm2.26dc  919  orandc  952  19.9ht  1694  ax11v2  1873  ax11v  1880  ax11ev  1881  equs5or  1883  nfsbxy  2002  nfsbxyt  2003  nfabdw  2411  eqvisset  2832  vtoclgf  2881  vtoclg1f  2882  eueq3dc  3000  mo2icl  3005  csbiegf  3191  un00  3567  vvin  3569  sneqr  3885  preqr1  3893  preq12b  3895  prel12  3896  nfopd  3921  ssex  4270  exmidundif  4343  iunpw  4626  nfimad  5135  dfrel2  5238  funsng  5427  cnvresid  5455  nffvd  5707  fnbrfvb  5741  funfvop  5821  acexmidlema  6076  tposf12  6540  supsnti  7346  pr2cv1  7542  exmidonfinlem  7546  sucpw1ne3  7592  sucpw1nel3  7593  recidnq  7761  ltaddnq  7775  ltadd1sr  8144  suplocsrlempr  8175  pncan3  8536  divcanap2  9013  ltp1  9177  ltm1  9179  recreclt  9233  nn0ind-raph  9768  2tnp1ge0ge0  10751  iswrdiz  11327  fsumcnv  12223  fprodcnv  12411  ef01bndlem  12542  sin01gt0  12548  cos01gt0  12549  ltoddhalfle  12679  bezoutlemnewy  12792  isprm5  12940  4sqlem12  13204  gzsumval2  13767  nmznsg  14069  gsump1  14241  tangtx  16031  gausslemma2dlem1a  16343  lgseisenlem4  16358  2lgslem3a  16378  2lgslem3b  16379  2lgslem3c  16380  2lgslem3d  16381  bdsepnft  17079  bdssex  17094  bj-inex  17099  bj-d0clsepcl  17117  bj-2inf  17130  bj-inf2vnlem2  17163