Theorem pm4.71i 388

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		pm4.71 386	. 2 |- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) )
3	1, 2	mpbi 144	1 |- ( ph <-> ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm4.24  392  anabs1  561  pm4.45  773  unidif0  4086  sucexb  4408  imadmrn  4886  dff1o2  5365  xpsnen  6708  dmaddpq  7180  dmmulpq  7181  eqreznegel  9399  xrnemnf  9557  xrnepnf  9558  elioopnf  9743  elioomnf  9744  elicopnf  9745  elxrge0  9754  isprm2  11787  bj-sucexg  13109