Theorem moeq3 1432

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 group(s):   φ,x   ψ,x   x,A   x,B   x,C

Proof of Theorem moeq3

Step	Hyp	Ref	Expression
1		cleq2 1110	. . . . . . 7 ⊢ (y = A → (x = y ↔ x = A))
2	1	anbi2d 468	. . . . . 6 ⊢ (y = A → ((φ ∧ x = y) ↔ (φ ∧ x = A)))
3		pm4.2i 149	. . . . . 6 ⊢ (y = A → ((¬ (φ ∨ ψ) ∧ x = B) ↔ (¬ (φ ∨ ψ) ∧ x = B)))
4		pm4.2i 149	. . . . . 6 ⊢ (y = A → ((ψ ∧ x = C) ↔ (ψ ∧ x = C)))
5	2, 3, 4	bi3ord 635	. . . . 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	bieudv 1013	. . . 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 1350	. . . . 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 1430	. . . 4 ⊢ ∃!x((φ ∧ x = y) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C))
12	6, 11	vtoclg 1383	. . 3 ⊢ (A ∈ V → ∃!x((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)))
13		eumo 1037	. . 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 12	. 2 ⊢ (A ∈ V → ∃*x((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)))
15		pm2.21 71	. . . . . . . . 9 ⊢ (¬ A ∈ V → (A ∈ V → x = y))
16		visset 1350	. . . . . . . . . 10 ⊢ x ∈ V
17		eleq1 1149	. . . . . . . . . 10 ⊢ (x = A → (x ∈ V ↔ A ∈ V))
18	16, 17	mpbii 168	. . . . . . . . 9 ⊢ (x = A → A ∈ V)
19	15, 18	syl5 22	. . . . . . . 8 ⊢ (¬ A ∈ V → (x = A → x = y))
20	19	anim2d 433	. . . . . . 7 ⊢ (¬ A ∈ V → ((φ ∧ x = A) → (φ ∧ x = y)))
21	20	orim1d 437	. . . . . 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 584	. . . . . 6 ⊢ (((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)) ↔ ((φ ∧ x = A) ∨ ((¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C))))
23		3orass 584	. . . . . 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 426	. . . . 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 943	. . . 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 1045	. . . 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 12	. . 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 110	1 ⊢ ∃*x((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C))

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   ∨ w3o 580  ∀wal 672   = weq 797  ∃!weu 1007  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  tz7.44lem1 2965

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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