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  8535  divcanap2  9012  ltp1  9176  ltm1  9178  recreclt  9232  nn0ind-raph  9767  2tnp1ge0ge0  10749  iswrdiz  11325  fsumcnv  12220  fprodcnv  12408  ef01bndlem  12539  sin01gt0  12545  cos01gt0  12546  ltoddhalfle  12676  bezoutlemnewy  12789  isprm5  12937  4sqlem12  13201  gzsumval2  13763  nmznsg  14065  gsump1  14206  tangtx  15989  gausslemma2dlem1a  16275  lgseisenlem4  16290  2lgslem3a  16310  2lgslem3b  16311  2lgslem3c  16312  2lgslem3d  16313  bdsepnft  17011  bdssex  17026  bj-inex  17031  bj-d0clsepcl  17049  bj-2inf  17062  bj-inf2vnlem2  17095