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  572  pm4.45  789  unidif0  4250  sucexb  4588  imadmrn  5077  dff1o2  5576  xpsnen  6976  dmaddpq  7562  dmmulpq  7563  eqreznegel  9805  xrnemnf  9969  xrnepnf  9970  elioopnf  10159  elioomnf  10160  elicopnf  10161  elxrge0  10170  dfrp2  10478  isprm2  12634  bj-sucexg  16243