Theorem eueq2 3672

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 143	. . . 4 ⊢ (𝜑 → ¬ ¬ 𝜑)
2		eueq2.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	eueqi 3671	. . . . 5 ⊢ ∃!𝑥 𝑥 = 𝐴
4		euanv 2651	. . . . . 6 ⊢ (∃!𝑥(𝜑 ∧ 𝑥 = 𝐴) ↔ (𝜑 ∧ ∃!𝑥 𝑥 = 𝐴))
5	4	biimpri 231	. . . . 5 ⊢ ((𝜑 ∧ ∃!𝑥 𝑥 = 𝐴) → ∃!𝑥(𝜑 ∧ 𝑥 = 𝐴))
6	3, 5	mpan2 703	. . . 4 ⊢ (𝜑 → ∃!𝑥(𝜑 ∧ 𝑥 = 𝐴))
7		euorv 2639	. . . 4 ⊢ ((¬ ¬ 𝜑 ∧ ∃!𝑥(𝜑 ∧ 𝑥 = 𝐴)) → ∃!𝑥(¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)))
8	1, 6, 7	syl2anc 595	. . 3 ⊢ (𝜑 → ∃!𝑥(¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)))
9		orcom 883	. . . . 5 ⊢ ((¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ ¬ 𝜑))
10	1	bianfd 543	. . . . . 6 ⊢ (𝜑 → (¬ 𝜑 ↔ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
11	10	orbi2d 928	. . . . 5 ⊢ (𝜑 → (((𝜑 ∧ 𝑥 = 𝐴) ∨ ¬ 𝜑) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
12	9, 11	bitrid 286	. . . 4 ⊢ (𝜑 → ((¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
13	12	eubidv 2613	. . 3 ⊢ (𝜑 → (∃!𝑥(¬ 𝜑 ∨ (𝜑 ∧ 𝑥 = 𝐴)) ↔ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
14	8, 13	mpbid 235	. 2 ⊢ (𝜑 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
15		eueq2.2	. . . . . 6 ⊢ 𝐵 ∈ V
16	15	eueqi 3671	. . . . 5 ⊢ ∃!𝑥 𝑥 = 𝐵
17		euanv 2651	. . . . . 6 ⊢ (∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵) ↔ (¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵))
18	17	biimpri 231	. . . . 5 ⊢ ((¬ 𝜑 ∧ ∃!𝑥 𝑥 = 𝐵) → ∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵))
19	16, 18	mpan2 703	. . . 4 ⊢ (¬ 𝜑 → ∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵))
20		euorv 2639	. . . 4 ⊢ ((¬ 𝜑 ∧ ∃!𝑥(¬ 𝜑 ∧ 𝑥 = 𝐵)) → ∃!𝑥(𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
21	19, 20	mpdan 699	. . 3 ⊢ (¬ 𝜑 → ∃!𝑥(𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
22		id 23	. . . . . 6 ⊢ (¬ 𝜑 → ¬ 𝜑)
23	22	bianfd 543	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 ↔ (𝜑 ∧ 𝑥 = 𝐴)))
24	23	orbi1d 929	. . . 4 ⊢ (¬ 𝜑 → ((𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
25	24	eubidv 2613	. . 3 ⊢ (¬ 𝜑 → (∃!𝑥(𝜑 ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)) ↔ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))))
26	21, 25	mpbid 235	. 2 ⊢ (¬ 𝜑 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵)))
27	14, 26	pm2.61i 184	1 ⊢ ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 = 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 400   ∨ wo 860   = wceq 1569   ∈ wcel 2142  ∃!weu 2595  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456

This theorem is used by:  bj-axseprep  37739