Theorem biadani 818

Description: Inference associated with biadan 817. (Contributed by BJ, 4-Mar-2023.)

Hypothesis

Ref	Expression
biadani.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
biadani	⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))

Proof of Theorem biadani

Step	Hyp	Ref	Expression
1		biadani.1	. 2 ⊢ (𝜑 → 𝜓)
2		biadan 817	. 2 ⊢ ((𝜑 → 𝜓) ↔ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒))))
3	1, 2	mpbi 232	1 ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  biadanii  820  elelb  34215