Theorem biadaniALT 833

Description: Alternate proof of biadani 832 not using biadan 831. (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		pm5.32 584	. 2 ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)))
2		biadani.1	. . . 4 ⊢ (𝜑 → 𝜓)
3	2	pm4.71ri 570	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑))
4	3	bibi1i 341	. 2 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) ↔ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)))
5	1, 4	bitr4i 281	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: (None)