Theorem moeq3dc 2864

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	⊢ ¬ (𝜑 ∧ 𝜓)

Assertion

Ref	Expression
moeq3dc	⊢ (DECID 𝜑 → (DECID 𝜓 → ∃*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 moeq3dc

Step	Hyp	Ref	Expression
1		moeq3dc.1	. . 3 ⊢ 𝐴 ∈ V
2		moeq3dc.2	. . 3 ⊢ 𝐵 ∈ V
3		moeq3dc.3	. . 3 ⊢ 𝐶 ∈ V
4		moeq3dc.4	. . 3 ⊢ ¬ (𝜑 ∧ 𝜓)
5	1, 2, 3, 4	eueq3dc 2862	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
6		eumo 2032	. 2 ⊢ (∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)) → ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))
7	5, 6	syl6 33	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  DECID wdc 820   ∨ w3o 962   = wceq 1332   ∈ wcel 1481  ∃!weu 2000  ∃*wmo 2001  Vcvv 2689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691

This theorem is referenced by: (None)