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	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
pm4.71i	⊢ (𝜑 ↔ (𝜑 ∧ 𝜓))

Proof of Theorem pm4.71i

Step	Hyp	Ref	Expression
1		pm4.71i.1	. 2 ⊢ (𝜑 → 𝜓)
2		pm4.71 389	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓)))
3	1, 2	mpbi 145	1 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜓))

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  4285  sucexb  4624  imadmrn  5116  dff1o2  5624  xpsnen  7085  dmaddpq  7710  dmmulpq  7711  eqreznegel  9964  xrnemnf  10129  xrnepnf  10130  elioopnf  10319  elioomnf  10320  elicopnf  10321  elxrge0  10330  dfrp2  10647  isprm2  12839  bj-sucexg  16818