Theorem moeq3 3609

Description: "At most one" property of equality (split into 3 cases). (The first two 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 2750	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
2	1	anbi2d 632	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜑 ∧ 𝑥 = 𝑦) ↔ (𝜑 ∧ 𝑥 = 𝐴)))
3		biidd 265	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ↔ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵)))
4		biidd 265	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ∧ 𝑥 = 𝐶) ↔ (𝜓 ∧ 𝑥 = 𝐶)))
5	2, 3, 4	3orbi123d 1436	. . . . 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 2587	. . . 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 3401	. . . . 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 3608	. . . 4 ⊢ ∃!𝑥((𝜑 ∧ 𝑥 = 𝑦) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))
12	6, 11	vtoclg 3470	. . 3 ⊢ (𝐴 ∈ V → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))
13		eumo 2579	. . 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 17	. 2 ⊢ (𝐴 ∈ V → ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))
15		eqvisset 3414	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
16		pm2.21 123	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∈ V → 𝑥 = 𝑦))
17	15, 16	syl5 34	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝑥 = 𝐴 → 𝑥 = 𝑦))
18	17	anim2d 615	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ((𝜑 ∧ 𝑥 = 𝐴) → (𝜑 ∧ 𝑥 = 𝑦)))
19	18	orim1d 965	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (((𝜑 ∧ 𝑥 = 𝐴) ∨ ((¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))) → ((𝜑 ∧ 𝑥 = 𝑦) ∨ ((¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))))
20		3orass 1091	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ ((¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
21		3orass 1091	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑦) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) ↔ ((𝜑 ∧ 𝑥 = 𝑦) ∨ ((¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
22	19, 20, 21	3imtr4g 299	. . . 4 ⊢ (¬ 𝐴 ∈ V → (((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) → ((𝜑 ∧ 𝑥 = 𝑦) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
23	22	alrimiv 1933	. . 3 ⊢ (¬ 𝐴 ∈ V → ∀𝑥(((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) → ((𝜑 ∧ 𝑥 = 𝑦) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
24		euimmo 2619	. . 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))))
25	23, 11, 24	mpisyl 21	. 2 ⊢ (¬ 𝐴 ∈ V → ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))
26	14, 25	pm2.61i 185	1 ⊢ ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 846   ∨ w3o 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2113  ∃*wmo 2538  ∃!weu 2569  Vcvv 3397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-tru 1545  df-ex 1787  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3399

This theorem is referenced by:  tz7.44lem1  8063