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Theorem moeq3 3535
 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 ¬ (𝜑𝜓)
Assertion
Ref Expression
moeq3 ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 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.
StepHypRef Expression
1 eqeq2 2782 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21anbi2d 614 . . . . . 6 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
3 biidd 252 . . . . . 6 (𝑦 = 𝐴 → ((¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ↔ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵)))
4 biidd 252 . . . . . 6 (𝑦 = 𝐴 → ((𝜓𝑥 = 𝐶) ↔ (𝜓𝑥 = 𝐶)))
52, 3, 43orbi123d 1546 . . . . 5 (𝑦 = 𝐴 → (((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
65eubidv 2638 . . . 4 (𝑦 = 𝐴 → (∃!𝑥((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) ↔ ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
7 vex 3354 . . . . 5 𝑦 ∈ V
8 moeq3.1 . . . . 5 𝐵 ∈ V
9 moeq3.2 . . . . 5 𝐶 ∈ V
10 moeq3.3 . . . . 5 ¬ (𝜑𝜓)
117, 8, 9, 10eueq3 3533 . . . 4 ∃!𝑥((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))
126, 11vtoclg 3417 . . 3 (𝐴 ∈ V → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)))
13 eumo 2647 . . 3 (∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) → ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)))
1412, 13syl 17 . 2 (𝐴 ∈ V → ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)))
15 eqvisset 3363 . . . . . . . 8 (𝑥 = 𝐴𝐴 ∈ V)
16 pm2.21 121 . . . . . . . 8 𝐴 ∈ V → (𝐴 ∈ V → 𝑥 = 𝑦))
1715, 16syl5 34 . . . . . . 7 𝐴 ∈ V → (𝑥 = 𝐴𝑥 = 𝑦))
1817anim2d 599 . . . . . 6 𝐴 ∈ V → ((𝜑𝑥 = 𝐴) → (𝜑𝑥 = 𝑦)))
1918orim1d 950 . . . . 5 𝐴 ∈ V → (((𝜑𝑥 = 𝐴) ∨ ((¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))) → ((𝜑𝑥 = 𝑦) ∨ ((¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)))))
20 3orass 1074 . . . . 5 (((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) ↔ ((𝜑𝑥 = 𝐴) ∨ ((¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
21 3orass 1074 . . . . 5 (((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) ↔ ((𝜑𝑥 = 𝑦) ∨ ((¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
2219, 20, 213imtr4g 285 . . . 4 𝐴 ∈ V → (((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) → ((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
2322alrimiv 2007 . . 3 𝐴 ∈ V → ∀𝑥(((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) → ((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
24 euimmo 2671 . . 3 (∀𝑥(((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) → ((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))) → (∃!𝑥((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) → ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
2523, 11, 24mpisyl 21 . 2 𝐴 ∈ V → ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)))
2614, 25pm2.61i 176 1 ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   ∨ wo 836   ∨ w3o 1070  ∀wal 1629   = wceq 1631   ∈ wcel 2145  ∃!weu 2618  ∃*wmo 2619  Vcvv 3351 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353 This theorem is referenced by:  tz7.44lem1  7654
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