Theorem bj-dfbi5 37424

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 884	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)) ↔ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∧ 𝜓)))
2		bj-dfbi4 37423	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
3		imor 867	. 2 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)) ↔ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∧ 𝜓)))
4	1, 2, 3	3bitr4i 306	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861

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  df-or 862

This theorem is used by:  bj-dfbi6  37425