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  562  pm4.45  774  unidif0  4140  sucexb  4468  imadmrn  4950  dff1o2  5431  xpsnen  6778  dmaddpq  7311  dmmulpq  7312  eqreznegel  9543  xrnemnf  9704  xrnepnf  9705  elioopnf  9894  elioomnf  9895  elicopnf  9896  elxrge0  9905  dfrp2  10189  isprm2  12028  bj-sucexg  13639