Theorem anidmdbi 575

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 574	. 2 ⊢ ((𝜓 ∧ 𝜓) ↔ 𝜓)
2	1	imbi2i 339	1 ⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401

This theorem is used by:  nanim  1527