Theorem pm4.71i 391

Description: Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.)

Hypothesis

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

Assertion

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

Proof of Theorem pm4.71i

Step	Hyp	Ref	Expression
1		pm4.71i.1	. 2 |- ( ph -> ps )
2		pm4.71 389	. 2 |- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) )
3	1, 2	mpbi 145	1 |- ( ph <-> ( 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-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm4.24  395  anabs1  572  pm4.45  784  unidif0  4162  sucexb  4490  imadmrn  4973  dff1o2  5458  xpsnen  6811  dmaddpq  7353  dmmulpq  7354  eqreznegel  9585  xrnemnf  9746  xrnepnf  9747  elioopnf  9936  elioomnf  9937  elicopnf  9938  elxrge0  9947  dfrp2  10232  isprm2  12082  bj-sucexg  14214