Theorem moeq3 3013

Description: "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.)

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

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2362	. . . . . . 7 ⊢ (y = A → (x = y ↔ x = A))
2	1	anbi2d 684	. . . . . 6 ⊢ (y = A → ((φ ∧ x = y) ↔ (φ ∧ x = A)))
3		biidd 228	. . . . . 6 ⊢ (y = A → ((¬ (φ ∨ ψ) ∧ x = B) ↔ (¬ (φ ∨ ψ) ∧ x = B)))
4		biidd 228	. . . . . 6 ⊢ (y = A → ((ψ ∧ x = C) ↔ (ψ ∧ x = C)))
5	2, 3, 4	3orbi123d 1251	. . . . 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 2212	. . . 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		vex 2862	. . . . 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 3011	. . . 4 ⊢ ∃!x((φ ∧ x = y) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C))
12	6, 11	vtoclg 2914	. . 3 ⊢ (A ∈ V → ∃!x((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)))
13		eumo 2244	. . 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 15	. 2 ⊢ (A ∈ V → ∃*x((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)))
15		vex 2862	. . . . . . . . 9 ⊢ x ∈ V
16		eleq1 2413	. . . . . . . . 9 ⊢ (x = A → (x ∈ V ↔ A ∈ V))
17	15, 16	mpbii 202	. . . . . . . 8 ⊢ (x = A → A ∈ V)
18		pm2.21 100	. . . . . . . 8 ⊢ (¬ A ∈ V → (A ∈ V → x = y))
19	17, 18	syl5 28	. . . . . . 7 ⊢ (¬ A ∈ V → (x = A → x = y))
20	19	anim2d 548	. . . . . 6 ⊢ (¬ A ∈ V → ((φ ∧ x = A) → (φ ∧ x = y)))
21	20	orim1d 812	. . . . 5 ⊢ (¬ A ∈ V → (((φ ∧ x = A) ∨ ((¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C))) → ((φ ∧ x = y) ∨ ((¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)))))
22		3orass 937	. . . . 5 ⊢ (((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)) ↔ ((φ ∧ x = A) ∨ ((¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C))))
23		3orass 937	. . . . 5 ⊢ (((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 261	. . . 4 ⊢ (¬ A ∈ V → (((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)) → ((φ ∧ x = y) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C))))
25	24	alrimiv 1631	. . 3 ⊢ (¬ 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 2253	. . 3 ⊢ (∀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, 11, 26	ee10 1376	. 2 ⊢ (¬ A ∈ V → ∃*x((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C)))
28	14, 27	pm2.61i 156	1 ⊢ ∃*x((φ ∧ x = A) ∨ (¬ (φ ∨ ψ) ∧ x = B) ∨ (ψ ∧ x = C))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   ∨ w3o 933  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  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-3or 935  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-nfc 2478  df-v 2861

This theorem is referenced by: (None)