Theorem iba 284

Description: Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) (Revised by NM, 24-Mar-2013.)

Assertion

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

Proof of Theorem iba

Step	Hyp	Ref	Expression
1		pm3.21 251	. 2 ⊢ (𝜑 → (𝜓 → (𝜓 ∧ 𝜑)))
2		simpl 102	. 2 ⊢ ((𝜓 ∧ 𝜑) → 𝜓)
3	1, 2	impbid1 130	1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  biantru  286  biantrud  288  ancrb  305  rbaibd  833  dedlem0a  875  fvopab6  5264  fressnfv  5350  tpostpos  5879  nnmword  6091  ltmpig  6437