Theorem dfbi3dc 1288

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 746	. . . 4 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
2		xordc 1283	. . . . 5 ⊢ (DECID 𝜑 → (DECID ¬ 𝜓 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑)))))
3	2	imp 115	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID ¬ 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
4	1, 3	sylan2 270	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
5		pm5.18dc 777	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
6	5	imp 115	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
7		notnotbdc 766	. . . . . 6 ⊢ (DECID 𝜓 → (𝜓 ↔ ¬ ¬ 𝜓))
8	7	anbi2d 437	. . . . 5 ⊢ (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ¬ ¬ 𝜓)))
9		ancom 253	. . . . . 6 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑))
10	9	a1i 9	. . . . 5 ⊢ (DECID 𝜓 → ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑)))
11	8, 10	orbi12d 707	. . . 4 ⊢ (DECID 𝜓 → (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
12	11	adantl 262	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
13	4, 6, 12	3bitr4d 209	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))))
14	13	ex 108	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743  df-xor 1267

This theorem is referenced by:  pm5.24dc  1289