Theorem biadaniALT 819

Description: Alternate proof of biadani 818 not using biadan 817. (Contributed by BJ, 4-Mar-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
biadani.1	⊢ (𝜑 → 𝜓)

Assertion

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

Proof of Theorem biadaniALT

Step	Hyp	Ref	Expression
1		pm5.32 576	. 2 ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)))
2		biadani.1	. . . 4 ⊢ (𝜑 → 𝜓)
3	2	pm4.71ri 563	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑))
4	3	bibi1i 341	. 2 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) ↔ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)))
5	1, 4	bitr4i 280	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: (None)