Theorem moeq3 3702

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 2833	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
2	1	anbi2d 630	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜑 ∧ 𝑥 = 𝑦) ↔ (𝜑 ∧ 𝑥 = 𝐴)))
3		biidd 264	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ↔ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵)))
4		biidd 264	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ∧ 𝑥 = 𝐶) ↔ (𝜓 ∧ 𝑥 = 𝐶)))
5	2, 3, 4	3orbi123d 1431	. . . . 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 2668	. . . 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 3497	. . . . 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 3701	. . . 4 ⊢ ∃!𝑥((𝜑 ∧ 𝑥 = 𝑦) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))
12	6, 11	vtoclg 3567	. . 3 ⊢ (𝐴 ∈ V → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))
13		eumo 2659	. . 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 3511	. . . . . . . 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 613	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → ((𝜑 ∧ 𝑥 = 𝐴) → (𝜑 ∧ 𝑥 = 𝑦)))
19	18	orim1d 962	. . . . 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 1086	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) ↔ ((𝜑 ∧ 𝑥 = 𝐴) ∨ ((¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
21		3orass 1086	. . . . 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 298	. . . 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 1924	. . 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 2696	. . 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 184	1 ⊢ ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   ∨ w3o 1082  ∀wal 1531   = wceq 1533   ∈ wcel 2110  ∃*wmo 2616  ∃!weu 2649  Vcvv 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3496

This theorem is referenced by:  tz7.44lem1  8035