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  576  biadani  616  orabs  822  prlem2  983  sb6  1935  2moswapdc  2170  exsnrex  3715  eliunxp  4875  asymref  5129  elxp4  5231  elxp5  5232  dffun9  5362  funcnv  5398  funcnv3  5399  f1ompt  5806  eufnfv  5895  dff1o6  5927  abexex  6297  dfoprab4  6364  tpostpos  6473  erovlem  6839  elixp2  6914  xpsnen  7048  ctssdccl  7370  ltbtwnnq  7696  enq0enq  7711  prnmaxl  7768  prnminu  7769  elznn0nn  9554  zrevaddcl  9591  qrevaddcl  9939  climreu  11937  isprm3  12770  isprm4  12771  xpscf  13510  tgval2  14862  eltg2b  14865  isms2  15265  2lgslem1b  15908