Theorem bj-dfbi6 31730

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

Assertion

Ref	Expression
bj-dfbi6	⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)))

Proof of Theorem bj-dfbi6

Step	Hyp	Ref	Expression
1		bj-dfbi5 31729	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))
2		id 22	. . . 4 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)) → ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))
3		animorr 505	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜓))
4	2, 3	impbid1 214	. . 3 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)) → ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)))
5		biimp 204	. . 3 ⊢ (((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)) → ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))
6	4, 5	impbii 198	. 2 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)))
7	1, 6	bitri 263	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383

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 196  df-or 384  df-an 385

This theorem is referenced by: (None)