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  5334  fvopab3g  5778  riota1a  6059  opabfi  7247  funisfsupp  7291  2omap  7318  ctssdccl  7451  2omotaplemap  7623  recmulnqg  7758  ltexprlemloc  7974  mul0eqap  9002  eluz2  9936  rpnegap  10097  elfz2  10428  zmodid2  10802  shftfib  11602  dvdsssfz1  12635  modremain  12712  ballotfilemdifcfz  13276  ctiunctlemudc  13377  issubg  14025  resgrpisgrp  14047  qusecsub  14184  issubrng  14556  issubrg  14578  txcnmpt  15423  reopnap  15696  ellimc3apf  15810  ralrals  17247  ralals  17253