Theorem ibar 301

Description: Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) (Revised by NM, 24-Mar-2013.)

Assertion

Ref	Expression
ibar	|- ( ph -> ( ps <-> ( ph /\ ps ) ) )

Proof of Theorem ibar

Step	Hyp	Ref	Expression
1		pm3.2 139	. 2 |- ( ph -> ( ps -> ( ph /\ ps ) ) )
2		simpr 110	. 2 |- ( ( ph /\ ps ) -> ps )
3	1, 2	impbid1 142	1 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  biantrur  303  biantrurd  305  anclb  319  pm5.42  320  pm5.32  457  anabs5  579  pm5.33  617  bianabs  619  annotanannot  680  baib  931  baibd  935  anxordi  1449  euan  2143  eueq3dc  3000  ifandc  3681  xpcom  5337  fvopab3g  5781  riota1a  6062  opabfi  7253  2omap  7324  ctssdccl  7457  2omotaplemap  7629  recmulnqg  7764  ltexprlemloc  7980  mul0eqap  9008  eluz2  9942  rpnegap  10103  elfz2  10434  zmodid2  10809  shftfib  11609  dvdsssfz1  12643  modremain  12720  ballotfilemdifcfz  13284  ctiunctlemudc  13385  issubg  14034  resgrpisgrp  14056  sscntz  14157  qusecsub  14224  issubrng  14615  issubrg  14637  txcnmpt  15516  reopnap  15789  ellimc3apf  15903  bpos  16337  ralrals  17373  ralals  17379