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Theorem moeq3dc 2740
Description: "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.)
Hypotheses
Ref Expression
moeq3dc.1 𝐴 ∈ V
moeq3dc.2 𝐵 ∈ V
moeq3dc.3 𝐶 ∈ V
moeq3dc.4 ¬ (𝜑𝜓)
Assertion
Ref Expression
moeq3dc (DECID 𝜑 → (DECID 𝜓 → ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
Distinct variable groups:   𝜑,𝑥   𝜓,𝑥   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem moeq3dc
StepHypRef Expression
1 moeq3dc.1 . . 3 𝐴 ∈ V
2 moeq3dc.2 . . 3 𝐵 ∈ V
3 moeq3dc.3 . . 3 𝐶 ∈ V
4 moeq3dc.4 . . 3 ¬ (𝜑𝜓)
51, 2, 3, 4eueq3dc 2738 . 2 (DECID 𝜑 → (DECID 𝜓 → ∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
6 eumo 1948 . 2 (∃!𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)) → ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶)))
75, 6syl6 33 1 (DECID 𝜑 → (DECID 𝜓 → ∃*𝑥((𝜑𝑥 = 𝐴) ∨ (¬ (𝜑𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓𝑥 = 𝐶))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639  DECID wdc 753  w3o 895   = wceq 1259  wcel 1409  ∃!weu 1916  ∃*wmo 1917  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by: (None)
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