Theorem eueq2dc 2857

Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)

Hypotheses

Ref	Expression
eueq2dc.1	⊢ 𝐴 ∈ V
eueq2dc.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
eueq2dc	⊢ (DECID 𝜑 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eueq2dc

Step	Hyp	Ref	Expression
1		df-dc 820	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		notnot 618	. . . . 5 ⊢ (𝜑 → ¬ ¬ 𝜑)
3		eueq2dc.1	. . . . . . 7 ⊢ 𝐴 ∈ V
4	3	eueq1 2856	. . . . . 6 ⊢ ∃!𝑥 𝑥 = 𝐴
5		euanv 2056	. . . . . . 7 ⊢ (∃!𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ (𝜑 ∧ ∃!𝑥 𝑥 = 𝐴))
6	5	biimpri 132	. . . . . 6 ⊢ ((𝜑 ∧ ∃!𝑥 𝑥 = 𝐴) → ∃!𝑥(𝜑 ∧ 𝑥 = 𝐴))
7	4, 6	mpan2 421	. . . . 5 ⊢ (𝜑 → ∃!𝑥(𝜑 ∧ 𝑥 = 𝐴))
8		euorv 2026	. . . . 5 ⊢ ((¬ ¬ 𝜑 ∧ ∃!𝑥(𝜑 ∧ 𝑥 = 𝐴)) → ∃!𝑥(¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)))
9	2, 7, 8	syl2anc 408	. . . 4 ⊢ (𝜑 → ∃!𝑥(¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)))
10		orcom 717	. . . . . 6 ⊢ ((¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ ¬ 𝜑))
11	2	bianfd 932	. . . . . . 7 ⊢ (𝜑 → (¬ 𝜑 ↔ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
12	11	orbi2d 779	. . . . . 6 ⊢ (𝜑 → (((𝜑 ∧ 𝑥 = 𝐴) ∨ ¬ 𝜑) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
13	10, 12	syl5bb 191	. . . . 5 ⊢ (𝜑 → ((¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
14	13	eubidv 2007	. . . 4 ⊢ (𝜑 → (∃!𝑥(¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)) ↔ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
15	9, 14	mpbid 146	. . 3 ⊢ (𝜑 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
16		eueq2dc.2	. . . . . . 7 ⊢ 𝐵 ∈ V
17	16	eueq1 2856	. . . . . 6 ⊢ ∃!𝑥 𝑥 = 𝐵
18		euanv 2056	. . . . . . 7 ⊢ (∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵) ↔ (¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵))
19	18	biimpri 132	. . . . . 6 ⊢ ((¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵) → ∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵))
20	17, 19	mpan2 421	. . . . 5 ⊢ (¬ 𝜑 → ∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵))
21		euorv 2026	. . . . 5 ⊢ ((¬ 𝜑 ∧ ∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵)) → ∃!𝑥(𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
22	20, 21	mpdan 417	. . . 4 ⊢ (¬ 𝜑 → ∃!𝑥(𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
23		id 19	. . . . . . 7 ⊢ (¬ 𝜑 → ¬ 𝜑)
24	23	bianfd 932	. . . . . 6 ⊢ (¬ 𝜑 → (𝜑 ↔ (𝜑 ∧ 𝑥 = 𝐴)))
25	24	orbi1d 780	. . . . 5 ⊢ (¬ 𝜑 → ((𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
26	25	eubidv 2007	. . . 4 ⊢ (¬ 𝜑 → (∃!𝑥(𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)) ↔ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
27	22, 26	mpbid 146	. . 3 ⊢ (¬ 𝜑 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
28	15, 27	jaoi 705	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
29	1, 28	sylbi 120	1 ⊢ (DECID 𝜑 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  DECID wdc 819   = wceq 1331   ∈ wcel 1480  ∃!weu 1999  Vcvv 2686

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-in1 603  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-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688

This theorem is referenced by: (None)