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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  biantrur  303  biantrurd  305  anclb  319  pm5.42  320  pm5.32  453  anabs5  575  pm5.33  613  bianabs  615  annotanannot  676  baib  927  baibd  931  anxordi  1445  euan  2136  eueq3dc  2981  ifandc  3650  xpcom  5290  fvopab3g  5728  riota1a  6002  opabfi  7175  funisfsupp  7216  ctssdccl  7370  2omotaplemap  7536  recmulnqg  7671  ltexprlemloc  7887  mul0eqap  8909  eluz2  9822  rpnegap  9982  elfz2  10312  zmodid2  10677  shftfib  11463  dvdsssfz1  12493  modremain  12570  ctiunctlemudc  13138  issubg  13840  resgrpisgrp  13862  qusecsub  13998  issubrng  14294  issubrg  14316  txcnmpt  15084  reopnap  15357  ellimc3apf  15471  2omap  16715