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  4272  exmidundif  4345  iunpw  4628  nfimad  5138  dfrel2  5241  funsng  5430  cnvresid  5458  nffvd  5710  fnbrfvb  5744  funfvop  5824  acexmidlema  6079  tposf12  6545  supsnti  7351  pr2cv1  7547  exmidonfinlem  7551  sucpw1ne3  7597  sucpw1nel3  7598  recidnq  7766  ltaddnq  7780  ltadd1sr  8149  suplocsrlempr  8180  pncan3  8541  divcanap2  9018  ltp1  9182  ltm1  9184  recreclt  9238  nn0ind-raph  9773  2tnp1ge0ge0  10756  iswrdiz  11332  fsumcnv  12228  fprodcnv  12416  ef01bndlem  12547  sin01gt0  12553  cos01gt0  12554  ltoddhalfle  12684  bezoutlemnewy  12797  isprm5  12945  4sqlem12  13209  gzsumval2  13772  nmznsg  14074  gsump1  14246  mvrf1fi  15219  tangtx  16082  gausslemma2dlem1a  16399  lgseisenlem4  16414  2lgslem3a  16434  2lgslem3b  16435  2lgslem3c  16436  2lgslem3d  16437  bdsepnft  17135  bdssex  17150  bj-inex  17155  bj-d0clsepcl  17173  bj-2inf  17186  bj-inf2vnlem2  17219