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Theorem moeq3 3349
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 2620 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21anbi2d 735 . . . . . 6 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
3 biidd 250 . . . . . 6 (𝑦 = 𝐴 → ((¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ↔ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵)))
4 biidd 250 . . . . . 6 (𝑦 = 𝐴 → ((𝜓𝑥 = 𝐶) ↔ (𝜓𝑥 = 𝐶)))
52, 3, 43orbi123d 1389 . . . . 5 (𝑦 = 𝐴 → (((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) ↔ ((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
65eubidv 2477 . . . 4 (𝑦 = 𝐴 → (∃!𝑥((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) ↔ ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
7 vex 3175 . . . . 5 𝑦 ∈ V
8 moeq3.1 . . . . 5 𝐵 ∈ V
9 moeq3.2 . . . . 5 𝐶 ∈ V
10 moeq3.3 . . . . 5 ¬ (𝜑𝜓)
117, 8, 9, 10eueq3 3347 . . . 4 ∃!𝑥((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))
126, 11vtoclg 3238 . . 3 (𝐴 ∈ V → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)))
13 eumo 2486 . . 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 3183 . . . . . . . 8 (𝑥 = 𝐴𝐴 ∈ V)
16 pm2.21 118 . . . . . . . 8 𝐴 ∈ V → (𝐴 ∈ V → 𝑥 = 𝑦))
1715, 16syl5 33 . . . . . . 7 𝐴 ∈ V → (𝑥 = 𝐴𝑥 = 𝑦))
1817anim2d 586 . . . . . 6 𝐴 ∈ V → ((𝜑𝑥 = 𝐴) → (𝜑𝑥 = 𝑦)))
1918orim1d 879 . . . . 5 𝐴 ∈ V → (((𝜑𝑥 = 𝐴) ∨ ((¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))) → ((𝜑𝑥 = 𝑦) ∨ ((¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)))))
20 3orass 1033 . . . . 5 (((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) ↔ ((𝜑𝑥 = 𝐴) ∨ ((¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
21 3orass 1033 . . . . 5 (((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) ↔ ((𝜑𝑥 = 𝑦) ∨ ((¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
2219, 20, 213imtr4g 283 . . . 4 𝐴 ∈ V → (((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) → ((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
2322alrimiv 1841 . . 3 𝐴 ∈ V → ∀𝑥(((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) → ((𝜑𝑥 = 𝑦) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
24 euimmo 2509 . . 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 174 1 ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382  w3o 1029  wal 1472   = wceq 1474  wcel 1976  ∃!weu 2457  ∃*wmo 2458  Vcvv 3172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-v 3174
This theorem is referenced by:  tz7.44lem1  7365
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