Theorem iba 300

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		pm3.21 264	. 2 ⊢ (𝜑 → (𝜓 → (𝜓 ∧ 𝜑)))
2		simpl 109	. 2 ⊢ ((𝜓 ∧ 𝜑) → 𝜓)
3	1, 2	impbid1 142	1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  biantru  302  biantrud  304  ancrb  322  rbaibd  925  dedlem0a  970  fvopab6  5628  fressnfv  5719  tpostpos  6283  nnmword  6537  unfiexmid  6935  ltmpig  7356  mul0eqap  8645  sup3exmid  8932  xrmaxiflemcom  11275