Theorem biadaniALT

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

	Ref	Expression
Hypothesis	biadani.1	⊢ ( 𝜑 → 𝜓 )
Assertion	biadaniALT	⊢ ( ( 𝜓 → ( 𝜑 ↔ 𝜒 ) ) ↔ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		biadani.1	⊢ ( 𝜑 → 𝜓 )
2		pm5.32	⊢ ( ( 𝜓 → ( 𝜑 ↔ 𝜒 ) ) ↔ ( ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜓 ∧ 𝜒 ) ) )
3	1	pm4.71ri	⊢ ( 𝜑 ↔ ( 𝜓 ∧ 𝜑 ) )
4	3	bibi1i	⊢ ( ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜓 ∧ 𝜑 ) ↔ ( 𝜓 ∧ 𝜒 ) ) )
5	2 4	bitr4i	⊢ ( ( 𝜓 → ( 𝜑 ↔ 𝜒 ) ) ↔ ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) )