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  912  orandc  945  19.9ht  1687  ax11v2  1866  ax11v  1873  ax11ev  1874  equs5or  1876  nfsbxy  1993  nfsbxyt  1994  nfabdw  2391  eqvisset  2810  vtoclgf  2859  vtoclg1f  2860  eueq3dc  2977  mo2icl  2982  csbiegf  3168  un00  3538  sneqr  3838  preqr1  3846  preq12b  3848  prel12  3849  nfopd  3874  ssex  4221  exmidundif  4290  iunpw  4571  nfimad  5077  dfrel2  5179  funsng  5367  cnvresid  5395  nffvd  5641  fnbrfvb  5674  funfvop  5749  acexmidlema  5998  tposf12  6421  supsnti  7183  pr2cv1  7379  exmidonfinlem  7382  sucpw1ne3  7428  sucpw1nel3  7429  recidnq  7591  ltaddnq  7605  ltadd1sr  7974  suplocsrlempr  8005  pncan3  8365  divcanap2  8838  ltp1  9002  ltm1  9004  recreclt  9058  nn0ind-raph  9575  2tnp1ge0ge0  10533  iswrdiz  11091  fsumcnv  11963  fprodcnv  12151  ef01bndlem  12282  sin01gt0  12288  cos01gt0  12289  ltoddhalfle  12419  bezoutlemnewy  12532  isprm5  12679  4sqlem12  12940  gsumval2  13445  nmznsg  13765  tangtx  15527  gausslemma2dlem1a  15752  lgseisenlem4  15767  2lgslem3a  15787  2lgslem3b  15788  2lgslem3c  15789  2lgslem3d  15790  bdsepnft  16305  bdssex  16320  bj-inex  16325  bj-d0clsepcl  16343  bj-2inf  16356  bj-inf2vnlem2  16389