Theorem moeq3 3383

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 2633	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
2	1	anbi2d 740	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜑 ∧ 𝑥 = 𝑦) ↔ (𝜑 ∧ 𝑥 = 𝐴)))
3		biidd 252	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ↔ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵)))
4		biidd 252	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ∧ 𝑥 = 𝐶) ↔ (𝜓 ∧ 𝑥 = 𝐶)))
5	2, 3, 4	3orbi123d 1398	. . . . 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 2490	. . . 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 3203	. . . . 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 3381	. . . 4 ⊢ ∃!𝑥((𝜑 ∧ 𝑥 = 𝑦) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))
12	6, 11	vtoclg 3266	. . 3 ⊢ (𝐴 ∈ V → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))
13		eumo 2499	. . 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 3211	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
16		pm2.21 120	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∈ V → 𝑥 = 𝑦))
17	15, 16	syl5 34	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝑥 = 𝐴 → 𝑥 = 𝑦))
18	17	anim2d 589	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ((𝜑 ∧ 𝑥 = 𝐴) → (𝜑 ∧ 𝑥 = 𝑦)))
19	18	orim1d 884	. . . . 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 1040	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ ((¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
21		3orass 1040	. . . . 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 285	. . . 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 1855	. . 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 2522	. . 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 176	1 ⊢ ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   ∨ w3o 1036  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∃!weu 2470  ∃*wmo 2471  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  tz7.44lem1  7501