Theorem eueq2 3380

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		notnot 136	. . . 4 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		eueq2.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	eueq1 3379	. . . . 5 ⊢ ∃!𝑥 𝑥 = 𝐴
4		euanv 2534	. . . . . 6 ⊢ (∃!𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ (𝜑 ∧ ∃!𝑥 𝑥 = 𝐴))
5	4	biimpri 218	. . . . 5 ⊢ ((𝜑 ∧ ∃!𝑥 𝑥 = 𝐴) → ∃!𝑥(𝜑 ∧ 𝑥 = 𝐴))
6	3, 5	mpan2 707	. . . 4 ⊢ (𝜑 → ∃!𝑥(𝜑 ∧ 𝑥 = 𝐴))
7		euorv 2513	. . . 4 ⊢ ((¬ ¬ 𝜑 ∧ ∃!𝑥(𝜑 ∧ 𝑥 = 𝐴)) → ∃!𝑥(¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)))
8	1, 6, 7	syl2anc 693	. . 3 ⊢ (𝜑 → ∃!𝑥(¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)))
9		orcom 402	. . . . 5 ⊢ ((¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ ¬ 𝜑))
10	1	bianfd 967	. . . . . 6 ⊢ (𝜑 → (¬ 𝜑 ↔ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
11	10	orbi2d 738	. . . . 5 ⊢ (𝜑 → (((𝜑 ∧ 𝑥 = 𝐴) ∨ ¬ 𝜑) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
12	9, 11	syl5bb 272	. . . 4 ⊢ (𝜑 → ((¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
13	12	eubidv 2490	. . 3 ⊢ (𝜑 → (∃!𝑥(¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)) ↔ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
14	8, 13	mpbid 222	. 2 ⊢ (𝜑 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
15		eueq2.2	. . . . . 6 ⊢ 𝐵 ∈ V
16	15	eueq1 3379	. . . . 5 ⊢ ∃!𝑥 𝑥 = 𝐵
17		euanv 2534	. . . . . 6 ⊢ (∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵) ↔ (¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵))
18	17	biimpri 218	. . . . 5 ⊢ ((¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵) → ∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵))
19	16, 18	mpan2 707	. . . 4 ⊢ (¬ 𝜑 → ∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵))
20		euorv 2513	. . . 4 ⊢ ((¬ 𝜑 ∧ ∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵)) → ∃!𝑥(𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
21	19, 20	mpdan 702	. . 3 ⊢ (¬ 𝜑 → ∃!𝑥(𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
22		id 22	. . . . . 6 ⊢ (¬ 𝜑 → ¬ 𝜑)
23	22	bianfd 967	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 ↔ (𝜑 ∧ 𝑥 = 𝐴)))
24	23	orbi1d 739	. . . 4 ⊢ (¬ 𝜑 → ((𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
25	24	eubidv 2490	. . 3 ⊢ (¬ 𝜑 → (∃!𝑥(𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)) ↔ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
26	21, 25	mpbid 222	. 2 ⊢ (¬ 𝜑 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
27	14, 26	pm2.61i 176	1 ⊢ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))

Syntax hints:  ¬ wn 3   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃!weu 2470  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)