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  3104  unineq  4268  fvopab6  7025  fressnfv  7155  tpostpos  8250  odi  8596  nnmword  8650  ltmpi  10923  maducoeval2  22583  mdbr2  32282  mdsl2i  32308  poimirlem26  37675  poimirlem27  37676  itg2addnclem  37700  itg2addnclem3  37702  prjspeclsp  42602  rmydioph  43005  expdioph  43014  dmafv2rnb  47225