Theorem eqifdc 3501

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 821	. . 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 3476	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐵)
5	3, 4	eqtrd 2170	. . . . . 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 3479	. . . . . . 7 ⊢ ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐶)
11	9, 10	eqtrd 2170	. . . . . 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 775	. . 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 3476	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → if(𝜑, 𝐵, 𝐶) = 𝐵)
19	16, 18	eqtr4d 2173	. . 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 3479	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → if(𝜑, 𝐵, 𝐶) = 𝐶)
23	20, 22	eqtr4d 2173	. . 3 ⊢ ((¬ 𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = if(𝜑, 𝐵, 𝐶))
24	19, 23	jaoi 705	. 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 697  DECID wdc 819   = wceq 1331  ifcif 3469

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 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-dc 820  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-if 3470

This theorem is referenced by:  fodjum  7011  xrmaxiflemcom  11011  subctctexmid  13185