Theorem ifnebibdc 3648

Description: The converse of ifbi 3623 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)

Assertion

Ref	Expression
ifnebibdc	⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓)))

Proof of Theorem ifnebibdc

Step	Hyp	Ref	Expression
1		eqifdc 3639	. . . 4 ⊢ (DECID 𝜓 → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∨ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵))))
2	1	3ad2ant2 1043	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∨ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵))))
3		ifnetruedc 3646	. . . . . . . . 9 ⊢ ((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
4	3	3expia 1229	. . . . . . . 8 ⊢ ((DECID 𝜑 ∧ 𝐴 ≠ 𝐵) → (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝜑))
5	4	3adant2 1040	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝜑))
6	5	adantld 278	. . . . . 6 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑))
7		simpl 109	. . . . . 6 ⊢ ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜓)
8	6, 7	jca2 308	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → (𝜑 ∧ 𝜓)))
9		pm5.1 603	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓))
10	8, 9	syl6 33	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → (𝜑 ↔ 𝜓)))
11		ifnefals 3647	. . . . . . . . 9 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)
12	11	ex 115	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = 𝐵 → ¬ 𝜑))
13	12	3ad2ant3 1044	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → (if(𝜑, 𝐴, 𝐵) = 𝐵 → ¬ 𝜑))
14	13	adantld 278	. . . . . 6 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → ((¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑))
15		simpl 109	. . . . . 6 ⊢ ((¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜓)
16	14, 15	jca2 308	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → ((¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → (¬ 𝜑 ∧ ¬ 𝜓)))
17		pm5.21 700	. . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ 𝜓))
18	16, 17	syl6 33	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → ((¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → (𝜑 ↔ 𝜓)))
19	10, 18	jaod 722	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → (((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∨ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → (𝜑 ↔ 𝜓)))
20	2, 19	sylbid 150	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) → (𝜑 ↔ 𝜓)))
21		ifbi 3623	. 2 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
22	20, 21	impbid1 142	1 ⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  DECID wdc 839   ∧ w3a 1002   = wceq 1395   ≠ wne 2400  ifcif 3602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ne 2401  df-if 3603

This theorem is referenced by:  nninfinf  10660