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  7345  pr2cv1  7541  exmidonfinlem  7545  sucpw1ne3  7591  sucpw1nel3  7592  recidnq  7760  ltaddnq  7774  ltadd1sr  8143  suplocsrlempr  8174  pncan3  8534  divcanap2  9010  ltp1  9174  ltm1  9176  recreclt  9230  nn0ind-raph  9763  2tnp1ge0ge0  10736  iswrdiz  11311  fsumcnv  12204  fprodcnv  12392  ef01bndlem  12523  sin01gt0  12529  cos01gt0  12530  ltoddhalfle  12660  bezoutlemnewy  12773  isprm5  12920  4sqlem12  13181  gzsumval2  13714  nmznsg  14016  gsump1  14157  tangtx  15939  gausslemma2dlem1a  16177  lgseisenlem4  16192  2lgslem3a  16212  2lgslem3b  16213  2lgslem3c  16214  2lgslem3d  16215  bdsepnft  16913  bdssex  16928  bj-inex  16933  bj-d0clsepcl  16951  bj-2inf  16964  bj-inf2vnlem2  16997