Theorem pm4.71i 389

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		pm4.71 387	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓)))
3	1, 2	mpbi 144	1 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜓))

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  393  anabs1  567  pm4.45  779  unidif0  4151  sucexb  4479  imadmrn  4961  dff1o2  5445  xpsnen  6795  dmaddpq  7328  dmmulpq  7329  eqreznegel  9560  xrnemnf  9721  xrnepnf  9722  elioopnf  9911  elioomnf  9912  elicopnf  9913  elxrge0  9922  dfrp2  10207  isprm2  12058  bj-sucexg  13879