Theorem iba 527

Description: Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.)

Assertion

Ref	Expression
iba	⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))

Proof of Theorem iba

Step	Hyp	Ref	Expression
1		pm3.21 471	. 2 ⊢ (𝜑 → (𝜓 → (𝜓 ∧ 𝜑)))
2		simpl 482	. 2 ⊢ ((𝜓 ∧ 𝜑) → 𝜓)
3	1, 2	impbid1 225	1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  ibar  528  biantru  529  biantrud  531  ancrb  547  pm5.54  1019  dedlem0a  1043  r19.29r  3093  unineq  4239  fvopab6  6964  fressnfv  7094  tpostpos  8179  odi  8497  nnmword  8551  ltmpi  10798  maducoeval2  22525  mdbr2  32244  mdsl2i  32270  poimirlem26  37646  poimirlem27  37647  itg2addnclem  37671  itg2addnclem3  37673  prjspeclsp  42605  rmydioph  43007  expdioph  43016  dmafv2rnb  47233