Theorem eueq2 3010

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

Hypotheses

Ref	Expression
eueq2.1	⊢ A ∈ V
eueq2.2	⊢ B ∈ V

Assertion

Ref	Expression
eueq2	⊢ ∃!x((φ ∧ x = A) ∨ (¬ φ ∧ x = B))

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

Proof of Theorem eueq2

Step	Hyp	Ref	Expression
1		notnot1 114	. . . 4 ⊢ (φ → ¬ ¬ φ)
2		eueq2.1	. . . . . 6 ⊢ A ∈ V
3	2	eueq1 3009	. . . . 5 ⊢ ∃!x x = A
4		euanv 2265	. . . . . 6 ⊢ (∃!x(φ ∧ x = A) ↔ (φ ∧ ∃!x x = A))
5	4	biimpri 197	. . . . 5 ⊢ ((φ ∧ ∃!x x = A) → ∃!x(φ ∧ x = A))
6	3, 5	mpan2 652	. . . 4 ⊢ (φ → ∃!x(φ ∧ x = A))
7		euorv 2232	. . . 4 ⊢ ((¬ ¬ φ ∧ ∃!x(φ ∧ x = A)) → ∃!x(¬ φ ∨ (φ ∧ x = A)))
8	1, 6, 7	syl2anc 642	. . 3 ⊢ (φ → ∃!x(¬ φ ∨ (φ ∧ x = A)))
9		orcom 376	. . . . 5 ⊢ ((¬ φ ∨ (φ ∧ x = A)) ↔ ((φ ∧ x = A) ∨ ¬ φ))
10	1	bianfd 892	. . . . . 6 ⊢ (φ → (¬ φ ↔ (¬ φ ∧ x = B)))
11	10	orbi2d 682	. . . . 5 ⊢ (φ → (((φ ∧ x = A) ∨ ¬ φ) ↔ ((φ ∧ x = A) ∨ (¬ φ ∧ x = B))))
12	9, 11	syl5bb 248	. . . 4 ⊢ (φ → ((¬ φ ∨ (φ ∧ x = A)) ↔ ((φ ∧ x = A) ∨ (¬ φ ∧ x = B))))
13	12	eubidv 2212	. . 3 ⊢ (φ → (∃!x(¬ φ ∨ (φ ∧ x = A)) ↔ ∃!x((φ ∧ x = A) ∨ (¬ φ ∧ x = B))))
14	8, 13	mpbid 201	. 2 ⊢ (φ → ∃!x((φ ∧ x = A) ∨ (¬ φ ∧ x = B)))
15		eueq2.2	. . . . . 6 ⊢ B ∈ V
16	15	eueq1 3009	. . . . 5 ⊢ ∃!x x = B
17		euanv 2265	. . . . . 6 ⊢ (∃!x(¬ φ ∧ x = B) ↔ (¬ φ ∧ ∃!x x = B))
18	17	biimpri 197	. . . . 5 ⊢ ((¬ φ ∧ ∃!x x = B) → ∃!x(¬ φ ∧ x = B))
19	16, 18	mpan2 652	. . . 4 ⊢ (¬ φ → ∃!x(¬ φ ∧ x = B))
20		euorv 2232	. . . 4 ⊢ ((¬ φ ∧ ∃!x(¬ φ ∧ x = B)) → ∃!x(φ ∨ (¬ φ ∧ x = B)))
21	19, 20	mpdan 649	. . 3 ⊢ (¬ φ → ∃!x(φ ∨ (¬ φ ∧ x = B)))
22		id 19	. . . . . 6 ⊢ (¬ φ → ¬ φ)
23	22	bianfd 892	. . . . 5 ⊢ (¬ φ → (φ ↔ (φ ∧ x = A)))
24	23	orbi1d 683	. . . 4 ⊢ (¬ φ → ((φ ∨ (¬ φ ∧ x = B)) ↔ ((φ ∧ x = A) ∨ (¬ φ ∧ x = B))))
25	24	eubidv 2212	. . 3 ⊢ (¬ φ → (∃!x(φ ∨ (¬ φ ∧ x = B)) ↔ ∃!x((φ ∧ x = A) ∨ (¬ φ ∧ x = B))))
26	21, 25	mpbid 201	. 2 ⊢ (¬ φ → ∃!x((φ ∧ x = A) ∨ (¬ φ ∧ x = B)))
27	14, 26	pm2.61i 156	1 ⊢ ∃!x((φ ∧ x = A) ∨ (¬ φ ∧ x = B))

Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by: (None)