Theorem bj-dfbi5 37029

Description: Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)

Assertion

Ref	Expression
bj-dfbi5	⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))

Proof of Theorem bj-dfbi5

Step	Hyp	Ref	Expression
1		orcom 883	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)) ↔ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∧ 𝜓)))
2		bj-dfbi4 37028	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
3		imor 866	. 2 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)) ↔ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∧ 𝜓)))
4	1, 2, 3	3bitr4i 306	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860

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 210  df-an 401  df-or 861

This theorem is referenced by:  bj-dfbi6  37030