Theorem eqifdc 3642

Description: Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)

Assertion

Ref	Expression
eqifdc	⊢ (DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))))

Proof of Theorem eqifdc

Step	Hyp	Ref	Expression
1		exmiddc 843	. . 3 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		simpr 110	. . . . . 6 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝜑)
3		simpl 109	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝐴 = if(𝜑, 𝐵, 𝐶))
4	2	iftrued 3612	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐵)
5	3, 4	eqtrd 2264	. . . . . 6 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝐴 = 𝐵)
6	2, 5	jca 306	. . . . 5 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → (𝜑 ∧ 𝐴 = 𝐵))
7	6	ex 115	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝜑 → (𝜑 ∧ 𝐴 = 𝐵)))
8		simpr 110	. . . . . 6 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → ¬ 𝜑)
9		simpl 109	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → 𝐴 = if(𝜑, 𝐵, 𝐶))
10	8	iffalsed 3615	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐶)
11	9, 10	eqtrd 2264	. . . . . 6 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → 𝐴 = 𝐶)
12	8, 11	jca 306	. . . . 5 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → (¬ 𝜑 ∧ 𝐴 = 𝐶))
13	12	ex 115	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (¬ 𝜑 → (¬ 𝜑 ∧ 𝐴 = 𝐶)))
14	7, 13	orim12d 793	. . 3 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → ((𝜑 ∨ ¬ 𝜑) → ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))))
15	1, 14	syl5com 29	. 2 ⊢ (DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) → ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))))
16		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
17		simpl 109	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜑)
18	17	iftrued 3612	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → if(𝜑, 𝐵, 𝐶) = 𝐵)
19	16, 18	eqtr4d 2267	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = if(𝜑, 𝐵, 𝐶))
20		simpr 110	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
21		simpl 109	. . . . 5 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → ¬ 𝜑)
22	21	iffalsed 3615	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → if(𝜑, 𝐵, 𝐶) = 𝐶)
23	20, 22	eqtr4d 2267	. . 3 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = if(𝜑, 𝐵, 𝐶))
24	19, 23	jaoi 723	. 2 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶)) → 𝐴 = if(𝜑, 𝐵, 𝐶))
25	15, 24	impbid1 142	1 ⊢ (DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  DECID wdc 841   = wceq 1397  ifcif 3605

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-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-if 3606

This theorem is referenced by:  ifnebibdc  3651  fodjum  7344  nninfwlporlemd  7370  xrmaxiflemcom  11809  gsumfzval  13473  subctctexmid  16601