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  791  unidif0  4257  sucexb  4595  imadmrn  5086  dff1o2  5588  xpsnen  7005  dmaddpq  7599  dmmulpq  7600  eqreznegel  9848  xrnemnf  10012  xrnepnf  10013  elioopnf  10202  elioomnf  10203  elicopnf  10204  elxrge0  10213  dfrp2  10523  isprm2  12690  bj-sucexg  16520