Theorem eqifdc 3511

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 822	. . 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 3486	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐵)
5	3, 4	eqtrd 2173	. . . . . 6 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝐴 = 𝐵)
6	2, 5	jca 304	. . . . 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 3489	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐶)
11	9, 10	eqtrd 2173	. . . . . 6 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → 𝐴 = 𝐶)
12	8, 11	jca 304	. . . . 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 776	. . 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 3486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → if(𝜑, 𝐵, 𝐶) = 𝐵)
19	16, 18	eqtr4d 2176	. . 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 3489	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → if(𝜑, 𝐵, 𝐶) = 𝐶)
23	20, 22	eqtr4d 2176	. . 3 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = if(𝜑, 𝐵, 𝐶))
24	19, 23	jaoi 706	. 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 698  DECID wdc 820   = wceq 1332  ifcif 3479

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-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-dc 821  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-if 3480

This theorem is referenced by:  fodjum  7026  xrmaxiflemcom  11050  subctctexmid  13369