Theorem biadani 612

Description: An implication implies to the equivalence of some implied equivalence and some other equivalence involving a conjunction. (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		pm5.32 453	. 2 ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)))
2		biadani.1	. . . 4 ⊢ (𝜑 → 𝜓)
3	2	pm4.71ri 392	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑))
4	3	bibi1i 228	. 2 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) ↔ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)))
5	1, 4	bitr4i 187	1 ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))

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:  biadanii  613