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Mirrors > Home > ILE Home > Th. List > moeq3dc | GIF version |
Description: "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.) |
Ref | Expression |
---|---|
moeq3dc.1 | ⊢ 𝐴 ∈ V |
moeq3dc.2 | ⊢ 𝐵 ∈ V |
moeq3dc.3 | ⊢ 𝐶 ∈ V |
moeq3dc.4 | ⊢ ¬ (𝜑 ∧ 𝜓) |
Ref | Expression |
---|---|
moeq3dc | ⊢ (DECID 𝜑 → (DECID 𝜓 → ∃*𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))) |
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 2858 | . 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → ∃!𝑥((𝜑 ∧ 𝑥 = 𝐴) ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝑥 = 𝐵) ∨ (𝜓 ∧ 𝑥 = 𝐶)))) |
6 | eumo 2031 | . 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)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 697 DECID wdc 819 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 ∃!weu 1999 ∃*wmo 2000 Vcvv 2686 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-v 2688 |
This theorem is referenced by: (None) |
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