Theorem anidmdbi 565

Description: Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)

Assertion

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

Proof of Theorem anidmdbi

Step	Hyp	Ref	Expression
1		anidm 564	. 2 ⊢ ((𝜓 ∧ 𝜓) ↔ 𝜓)
2	1	imbi2i 335	1 ⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ 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 206  df-an 396

This theorem is referenced by:  nanim  1490