Theorem moeq3dc 2928

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 2926	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶))))
6		eumo 2070	. 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 104   ∨ wo 709  DECID wdc 835   ∨ w3o 979   = wceq 1364  ∃!weu 2038  ∃*wmo 2039   ∈ wcel 2160  Vcvv 2752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754

This theorem is referenced by: (None)