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  7319  ctssdccl  7452  2omotaplemap  7624  recmulnqg  7759  ltexprlemloc  7975  mul0eqap  9003  eluz2  9937  rpnegap  10098  elfz2  10429  zmodid2  10804  shftfib  11604  dvdsssfz1  12638  modremain  12715  ballotfilemdifcfz  13279  ctiunctlemudc  13380  issubg  14029  resgrpisgrp  14051  sscntz  14152  qusecsub  14219  issubrng  14591  issubrg  14613  txcnmpt  15465  reopnap  15738  ellimc3apf  15852  bpos  16281  ralrals  17316  ralals  17322