Theorem dfbi3OLD 995

Description: Obsolete proof of dfbi3 994 as of 29-Oct-2021. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
dfbi3OLD	⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))

Proof of Theorem dfbi3OLD

Step	Hyp	Ref	Expression
1		xor 935	. 2 ⊢ (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑)))
2		pm5.18 371	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
3		notnotb 304	. . . 4 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
4	3	anbi2i 730	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ¬ ¬ 𝜓))
5		ancom 466	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑))
6	4, 5	orbi12i 543	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑)))
7	1, 2, 6	3bitr4i 292	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by: (None)