Theorem moeq3 1912

Description: "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.)

Hypotheses

Ref	Expression
moeq3.1	⊢ B ∈ V
moeq3.2	⊢ C ∈ V
moeq3.3	⊢ ¬ (φ ⋀ ψ)

Assertion

Ref	Expression
moeq3	⊢ ∃*x((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))

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

Proof of Theorem moeq3

Step	Hyp	Ref	Expression
1		eqeq2 1476	. . . . . . 7 ⊢ (y = A → (x = y ↔ x = A))
2	1	anbi2d 614	. . . . . 6 ⊢ (y = A → ((φ ⋀ x = y) ↔ (φ ⋀ x = A)))
3		pm4.2i 171	. . . . . 6 ⊢ (y = A → ((¬ (φ ⋁ ψ) ⋀ x = B) ↔ (¬ (φ ⋁ ψ) ⋀ x = B)))
4		pm4.2i 171	. . . . . 6 ⊢ (y = A → ((ψ ⋀ x = C) ↔ (ψ ⋀ x = C)))
5	2, 3, 4	3orbi123d 889	. . . . 5 ⊢ (y = A → (((φ ⋀ x = y) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)) ↔ ((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))))
6	5	eubidv 1379	. . . 4 ⊢ (y = A → (∃!x((φ ⋀ x = y) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)) ↔ ∃!x((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))))
7		visset 1804	. . . . 5 ⊢ y ∈ V
8		moeq3.1	. . . . 5 ⊢ B ∈ V
9		moeq3.2	. . . . 5 ⊢ C ∈ V
10		moeq3.3	. . . . 5 ⊢ ¬ (φ ⋀ ψ)
11	7, 8, 9, 10	eueq3 1910	. . . 4 ⊢ ∃!x((φ ⋀ x = y) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))
12	6, 11	vtoclg 1838	. . 3 ⊢ (A ∈ V → ∃!x((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)))
13		eumo 1404	. . 3 ⊢ (∃!x((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)) → ∃*x((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)))
14	12, 13	syl 10	. 2 ⊢ (A ∈ V → ∃*x((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)))
15		pm2.21 76	. . . . . . . . 9 ⊢ (¬ A ∈ V → (A ∈ V → x = y))
16		visset 1804	. . . . . . . . . 10 ⊢ x ∈ V
17		eleq1 1526	. . . . . . . . . 10 ⊢ (x = A → (x ∈ V ↔ A ∈ V))
18	16, 17	mpbii 193	. . . . . . . . 9 ⊢ (x = A → A ∈ V)
19	15, 18	syl5 21	. . . . . . . 8 ⊢ (¬ A ∈ V → (x = A → x = y))
20	19	anim2d 559	. . . . . . 7 ⊢ (¬ A ∈ V → ((φ ⋀ x = A) → (φ ⋀ x = y)))
21	20	orim1d 564	. . . . . 6 ⊢ (¬ A ∈ V → (((φ ⋀ x = A) ⋁ ((¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))) → ((φ ⋀ x = y) ⋁ ((¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)))))
22		3orass 776	. . . . . 6 ⊢ (((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)) ↔ ((φ ⋀ x = A) ⋁ ((¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))))
23		3orass 776	. . . . . 6 ⊢ (((φ ⋀ x = y) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)) ↔ ((φ ⋀ x = y) ⋁ ((¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))))
24	21, 22, 23	3imtr4g 551	. . . . 5 ⊢ (¬ A ∈ V → (((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)) → ((φ ⋀ x = y) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))))
25	24	19.21aiv 1281	. . . 4 ⊢ (¬ A ∈ V → ∀x(((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)) → ((φ ⋀ x = y) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))))
26		euimmo 1413	. . . 4 ⊢ (∀x(((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)) → ((φ ⋀ x = y) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))) → (∃!x((φ ⋀ x = y) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)) → ∃*x((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))))
27	25, 26	syl 10	. . 3 ⊢ (¬ A ∈ V → (∃!x((φ ⋀ x = y) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)) → ∃*x((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))))
28	11, 27	mpi 44	. 2 ⊢ (¬ A ∈ V → ∃*x((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C)))
29	14, 28	pm2.61i 126	1 ⊢ ∃*x((φ ⋀ x = A) ⋁ (¬ (φ ⋁ ψ) ⋀ x = B) ⋁ (ψ ⋀ x = C))

Syntax hints:  ¬ wn 2   → wi 3   ⋁ wo 222   ⋀ wa 223   ⋁ w3o 772  ∀wal 951   = wceq 953   ∈ wcel 955  ∃!weu 1373  ∃*wmo 1374  Vcvv 1802

This theorem is referenced by:  tz7.44lem1 3912

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

Copyright terms: Public domain