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  574  pm4.45  792  unidif0  4263  sucexb  4601  imadmrn  5092  dff1o2  5597  xpsnen  7048  dmaddpq  7659  dmmulpq  7660  eqreznegel  9909  xrnemnf  10073  xrnepnf  10074  elioopnf  10263  elioomnf  10264  elicopnf  10265  elxrge0  10274  dfrp2  10586  isprm2  12769  bj-sucexg  16638