Theorem biadani 832

Description: Inference associated with biadan 831. (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 831	. 2 ⊢ ((𝜑 → 𝜓) ↔ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒))))
3	1, 2	mpbi 233	1 ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

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 402

This theorem is used by:  biadanii  834  elelb  37591