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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 (𝜑𝜓)
Assertion
Ref Expression
pm4.71i (𝜑 ↔ (𝜑𝜓))

Proof of Theorem pm4.71i
StepHypRef Expression
1 pm4.71i.1 . 2 (𝜑𝜓)
2 pm4.71 387 . 2 ((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
31, 2mpbi 144 1 (𝜑 ↔ (𝜑𝜓))
Colors of variables: wff set class
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  4099  sucexb  4421  imadmrn  4899  dff1o2  5380  xpsnen  6723  dmaddpq  7211  dmmulpq  7212  eqreznegel  9433  xrnemnf  9594  xrnepnf  9595  elioopnf  9780  elioomnf  9781  elicopnf  9782  elxrge0  9791  isprm2  11834  bj-sucexg  13291
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