Theorem pm4.71ri 392

Description: Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 1-Dec-2003.)

Hypothesis

Ref	Expression
pm4.71ri.1	|- ( ph -> ps )

Assertion

Ref	Expression
pm4.71ri	|- ( ph <-> ( ps /\ ph ) )

Proof of Theorem pm4.71ri

Step	Hyp	Ref	Expression
1		pm4.71ri.1	. 2 |- ( ph -> ps )
2		pm4.71r 390	. 2 |- ( ( ph -> ps ) <-> ( ph <-> ( ps /\ ph ) ) )
3	1, 2	mpbi 145	1 |- ( ph <-> ( ps /\ ph ) )

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

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  biadan2  456  anabs7  574  biadani  612  orabs  815  prlem2  976  sb6  1901  2moswapdc  2135  exsnrex  3665  eliunxp  4806  asymref  5056  elxp4  5158  elxp5  5159  dffun9  5288  funcnv  5320  funcnv3  5321  f1ompt  5716  eufnfv  5796  dff1o6  5826  abexex  6192  dfoprab4  6259  tpostpos  6331  erovlem  6695  elixp2  6770  xpsnen  6889  ctssdccl  7186  ltbtwnnq  7500  enq0enq  7515  prnmaxl  7572  prnminu  7573  elznn0nn  9357  zrevaddcl  9393  qrevaddcl  9735  climreu  11479  isprm3  12311  isprm4  12312  xpscf  13049  tgval2  14371  eltg2b  14374  isms2  14774  2lgslem1b  15414