Theorem moeq3 3673

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 2775	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
2	1	anbi2d 642	. . . . . 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 1464	. . . . 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 2614	. . . 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 3458	. . . . 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 3672	. . . 4 ⊢ ∃!𝑥((𝜑 ∧ 𝑥 = 𝑦) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))
12	6, 11	vtoclg 3521	. . 3 ⊢ (𝐴 ∈ V → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))
13		eumo 2606	. . 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 18	. 2 ⊢ (𝐴 ∈ V → ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))
15		eqvisset 3474	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
16		pm2.21 124	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∈ V → 𝑥 = 𝑦))
17	15, 16	syl5 35	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝑥 = 𝐴 → 𝑥 = 𝑦))
18	17	anim2d 624	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ((𝜑 ∧ 𝑥 = 𝐴) → (𝜑 ∧ 𝑥 = 𝑦)))
19	18	orim1d 981	. . . . 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 1106	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ ((¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
21		3orass 1106	. . . . 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 1962	. . 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 2644	. . 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 22	. 2 ⊢ (¬ 𝐴 ∈ V → ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))
26	14, 25	pm2.61i 184	1 ⊢ ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102  ∀wal 1568   = wceq 1570   ∈ wcel 2147  ∃*wmo 2565  ∃!weu 2596  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-tru 1573  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456

This theorem is used by:  tz7.44lem1  8438