Theorem eqifdc 3453

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 788	. . 3 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		simpr 109	. . . . . 6 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝜑)
3		simpl 108	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝐴 = if(𝜑, 𝐵, 𝐶))
4	2	iftrued 3428	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐵)
5	3, 4	eqtrd 2132	. . . . . 6 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝐴 = 𝐵)
6	2, 5	jca 302	. . . . 5 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → (𝜑 ∧ 𝐴 = 𝐵))
7	6	ex 114	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝜑 → (𝜑 ∧ 𝐴 = 𝐵)))
8		simpr 109	. . . . . 6 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → ¬ 𝜑)
9		simpl 108	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → 𝐴 = if(𝜑, 𝐵, 𝐶))
10	8	iffalsed 3431	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐶)
11	9, 10	eqtrd 2132	. . . . . 6 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → 𝐴 = 𝐶)
12	8, 11	jca 302	. . . . 5 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → (¬ 𝜑 ∧ 𝐴 = 𝐶))
13	12	ex 114	. . . 4 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (¬ 𝜑 → (¬ 𝜑 ∧ 𝐴 = 𝐶)))
14	7, 13	orim12d 741	. . 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 109	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
17		simpl 108	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜑)
18	17	iftrued 3428	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → if(𝜑, 𝐵, 𝐶) = 𝐵)
19	16, 18	eqtr4d 2135	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = if(𝜑, 𝐵, 𝐶))
20		simpr 109	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
21		simpl 108	. . . . 5 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → ¬ 𝜑)
22	21	iffalsed 3431	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → if(𝜑, 𝐵, 𝐶) = 𝐶)
23	20, 22	eqtr4d 2135	. . 3 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = if(𝜑, 𝐵, 𝐶))
24	19, 23	jaoi 677	. 2 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶)) → 𝐴 = if(𝜑, 𝐵, 𝐶))
25	15, 24	impbid1 141	1 ⊢ (DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 670  DECID wdc 786   = wceq 1299  ifcif 3421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-dc 787  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-if 3422

This theorem is referenced by:  fodjum  6930  xrmaxiflemcom  10857