Theorem dfbi3 1049

Description: An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) (Proof shortened by NM, 29-Oct-2021.)

Assertion

Ref	Expression
dfbi3	⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))

Proof of Theorem dfbi3

Step	Hyp	Ref	Expression
1		con34b 316	. . 3 ⊢ ((𝜓 → 𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
2	1	anbi2i 623	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
3		dfbi2 474	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
4		cases2 1047	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → ¬ 𝜓)))
5	2, 3, 4	3bitr4i 303	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847

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 207  df-an 396  df-or 848

This theorem is referenced by:  pm5.24  1050  nanbi  1499  raaan2  4520  2reu4lem  4521  ifbi  4547  sqf11  27183  bj-dfbi4  36575