Theorem iba 298

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		pm3.21 262	. 2 ⊢ (𝜑 → (𝜓 → (𝜓 ∧ 𝜑)))
2		simpl 108	. 2 ⊢ ((𝜓 ∧ 𝜑) → 𝜓)
3	1, 2	impbid1 141	1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))

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:  biantru  300  biantrud  302  ancrb  320  rbaibd  909  dedlem0a  952  fvopab6  5517  fressnfv  5607  tpostpos  6161  nnmword  6414  unfiexmid  6806  ltmpig  7147  mul0eqap  8431  sup3exmid  8715  xrmaxiflemcom  11018