ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  euotd GIF version

Theorem euotd 4037
Description: Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)
Hypotheses
Ref Expression
euotd.1 (𝜑𝐴 ∈ V)
euotd.2 (𝜑𝐵 ∈ V)
euotd.3 (𝜑𝐶 ∈ V)
euotd.4 (𝜑 → (𝜓 ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)))
Assertion
Ref Expression
euotd (𝜑 → ∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑥,𝐴   𝐵,𝑎,𝑏,𝑐,𝑥   𝐶,𝑎,𝑏,𝑐,𝑥   𝜑,𝑎,𝑏,𝑐,𝑥
Allowed substitution hints:   𝜓(𝑥,𝑎,𝑏,𝑐)

Proof of Theorem euotd
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 euotd.1 . . . 4 (𝜑𝐴 ∈ V)
2 euotd.2 . . . 4 (𝜑𝐵 ∈ V)
3 euotd.3 . . . 4 (𝜑𝐶 ∈ V)
4 otexg 4013 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
51, 2, 3, 4syl3anc 1170 . . 3 (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
6 euotd.4 . . . . . . . . . . . . 13 (𝜑 → (𝜓 ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)))
76biimpa 290 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
8 vex 2613 . . . . . . . . . . . . 13 𝑎 ∈ V
9 vex 2613 . . . . . . . . . . . . 13 𝑏 ∈ V
10 vex 2613 . . . . . . . . . . . . 13 𝑐 ∈ V
118, 9, 10otth 4025 . . . . . . . . . . . 12 (⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩ ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
127, 11sylibr 132 . . . . . . . . . . 11 ((𝜑𝜓) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
1312eqeq2d 2094 . . . . . . . . . 10 ((𝜑𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1413biimpd 142 . . . . . . . . 9 ((𝜑𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1514impancom 256 . . . . . . . 8 ((𝜑𝑥 = ⟨𝑎, 𝑏, 𝑐⟩) → (𝜓𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1615expimpd 355 . . . . . . 7 (𝜑 → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1716exlimdv 1742 . . . . . 6 (𝜑 → (∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1817exlimdvv 1820 . . . . 5 (𝜑 → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
19 tru 1289 . . . . . . . . . . 11
202adantr 270 . . . . . . . . . . . . 13 ((𝜑𝑎 = 𝐴) → 𝐵 ∈ V)
213ad2antrr 472 . . . . . . . . . . . . . 14 (((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝐶 ∈ V)
22 simpr 108 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
2322, 11sylibr 132 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
2423eqcomd 2088 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩)
256biimpar 291 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → 𝜓)
2624, 25jca 300 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
27 a1tru 1301 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⊤)
2826, 272thd 173 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
29283anassrs 1161 . . . . . . . . . . . . . 14 ((((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3021, 29sbcied 2859 . . . . . . . . . . . . 13 (((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3120, 30sbcied 2859 . . . . . . . . . . . 12 ((𝜑𝑎 = 𝐴) → ([𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
321, 31sbcied 2859 . . . . . . . . . . 11 (𝜑 → ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3319, 32mpbiri 166 . . . . . . . . . 10 (𝜑[𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
3433spesbcd 2909 . . . . . . . . 9 (𝜑 → ∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
35 nfcv 2223 . . . . . . . . . 10 𝑏𝐵
36 nfsbc1v 2842 . . . . . . . . . . 11 𝑏[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
3736nfex 1569 . . . . . . . . . 10 𝑏𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
38 sbceq1a 2833 . . . . . . . . . . 11 (𝑏 = 𝐵 → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
3938exbidv 1748 . . . . . . . . . 10 (𝑏 = 𝐵 → (∃𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
4035, 37, 39spcegf 2690 . . . . . . . . 9 (𝐵 ∈ V → (∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → ∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
412, 34, 40sylc 61 . . . . . . . 8 (𝜑 → ∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
42 nfcv 2223 . . . . . . . . 9 𝑐𝐶
43 nfsbc1v 2842 . . . . . . . . . . 11 𝑐[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
4443nfex 1569 . . . . . . . . . 10 𝑐𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
4544nfex 1569 . . . . . . . . 9 𝑐𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
46 sbceq1a 2833 . . . . . . . . . 10 (𝑐 = 𝐶 → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
47462exbidv 1791 . . . . . . . . 9 (𝑐 = 𝐶 → (∃𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
4842, 45, 47spcegf 2690 . . . . . . . 8 (𝐶 ∈ V → (∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → ∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
493, 41, 48sylc 61 . . . . . . 7 (𝜑 → ∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
50 excom13 1620 . . . . . . 7 (∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
5149, 50sylib 120 . . . . . 6 (𝜑 → ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
52 eqeq1 2089 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩))
5352anbi1d 453 . . . . . . 7 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
54533exbidv 1792 . . . . . 6 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
5551, 54syl5ibrcom 155 . . . . 5 (𝜑 → (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
5618, 55impbid 127 . . . 4 (𝜑 → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
5756alrimiv 1797 . . 3 (𝜑 → ∀𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
58 eqeq2 2092 . . . . . 6 (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = 𝑦𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
5958bibi2d 230 . . . . 5 (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → ((∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦) ↔ (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩)))
6059albidv 1747 . . . 4 (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → (∀𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩)))
6160spcegv 2695 . . 3 (⟨𝐴, 𝐵, 𝐶⟩ ∈ V → (∀𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩) → ∃𝑦𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦)))
625, 57, 61sylc 61 . 2 (𝜑 → ∃𝑦𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
63 df-eu 1946 . 2 (∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑦𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
6462, 63sylibr 132 1 (𝜑 → ∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920  wal 1283   = wceq 1285  wtru 1286  wex 1422  wcel 1434  ∃!weu 1943  Vcvv 2610  [wsbc 2824  cotp 3420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-ot 3426
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator