Theorem dfbi3dc 1392

Description: An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.)

Assertion

Ref	Expression
dfbi3dc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))))

Proof of Theorem dfbi3dc

Step	Hyp	Ref	Expression
1		dcn 837	. . . 4 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
2		xordc 1387	. . . . 5 ⊢ (DECID 𝜑 → (DECID ¬ 𝜓 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑)))))
3	2	imp 123	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID ¬ 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
4	1, 3	sylan2 284	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
5		pm5.18dc 878	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
6	5	imp 123	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
7		notnotbdc 867	. . . . . 6 ⊢ (DECID 𝜓 → (𝜓 ↔ ¬ ¬ 𝜓))
8	7	anbi2d 461	. . . . 5 ⊢ (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ¬ ¬ 𝜓)))
9		ancom 264	. . . . . 6 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑))
10	9	a1i 9	. . . . 5 ⊢ (DECID 𝜓 → ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑)))
11	8, 10	orbi12d 788	. . . 4 ⊢ (DECID 𝜓 → (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
12	11	adantl 275	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
13	4, 6, 12	3bitr4d 219	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))))
14	13	ex 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-xor 1371

This theorem is referenced by:  pm5.24dc  1393