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  4169  sucexb  4498  imadmrn  4982  dff1o2  5468  xpsnen  6824  dmaddpq  7381  dmmulpq  7382  eqreznegel  9617  xrnemnf  9780  xrnepnf  9781  elioopnf  9970  elioomnf  9971  elicopnf  9972  elxrge0  9981  dfrp2  10267  isprm2  12120  bj-sucexg  14814