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 )

Syntax hints:   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm2.26dc  914  orandc  947  19.9ht  1689  ax11v2  1868  ax11v  1875  ax11ev  1876  equs5or  1878  nfsbxy  1995  nfsbxyt  1996  nfabdw  2393  eqvisset  2813  vtoclgf  2862  vtoclg1f  2863  eueq3dc  2980  mo2icl  2985  csbiegf  3171  un00  3541  sneqr  3843  preqr1  3851  preq12b  3853  prel12  3854  nfopd  3879  ssex  4226  exmidundif  4296  iunpw  4577  nfimad  5085  dfrel2  5187  funsng  5376  cnvresid  5404  nffvd  5651  fnbrfvb  5684  funfvop  5759  acexmidlema  6008  tposf12  6434  supsnti  7203  pr2cv1  7399  exmidonfinlem  7403  sucpw1ne3  7449  sucpw1nel3  7450  recidnq  7612  ltaddnq  7626  ltadd1sr  7995  suplocsrlempr  8026  pncan3  8386  divcanap2  8859  ltp1  9023  ltm1  9025  recreclt  9079  nn0ind-raph  9596  2tnp1ge0ge0  10560  iswrdiz  11119  fsumcnv  11997  fprodcnv  12185  ef01bndlem  12316  sin01gt0  12322  cos01gt0  12323  ltoddhalfle  12453  bezoutlemnewy  12566  isprm5  12713  4sqlem12  12974  gsumval2  13479  nmznsg  13799  tangtx  15561  gausslemma2dlem1a  15786  lgseisenlem4  15801  2lgslem3a  15821  2lgslem3b  15822  2lgslem3c  15823  2lgslem3d  15824  bdsepnft  16482  bdssex  16497  bj-inex  16502  bj-d0clsepcl  16520  bj-2inf  16533  bj-inf2vnlem2  16566