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Theorem dfbi3OLD 994
 Description: Obsolete proof of dfbi3 993 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 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))

Proof of Theorem dfbi3OLD
StepHypRef Expression
1 xor 934 . 2 (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑)))
2 pm5.18 371 . 2 ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
3 notnotb 304 . . . 4 (𝜓 ↔ ¬ ¬ 𝜓)
43anbi2i 729 . . 3 ((𝜑𝜓) ↔ (𝜑 ∧ ¬ ¬ 𝜓))
5 ancom 466 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑))
64, 5orbi12i 543 . 2 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑)))
71, 2, 63bitr4i 292 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
 Colors of variables: wff setvar class 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)
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