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Theorem euotd 3991
 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 3967 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
51, 2, 3, 4syl3anc 1135 . . 3 (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
6 euotd.4 . . . . . . . . . . . . 13 (𝜑 → (𝜓 ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)))
76biimpa 280 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
8 vex 2560 . . . . . . . . . . . . 13 𝑎 ∈ V
9 vex 2560 . . . . . . . . . . . . 13 𝑏 ∈ V
10 vex 2560 . . . . . . . . . . . . 13 𝑐 ∈ V
118, 9, 10otth 3979 . . . . . . . . . . . 12 (⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩ ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
127, 11sylibr 137 . . . . . . . . . . 11 ((𝜑𝜓) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
1312eqeq2d 2051 . . . . . . . . . 10 ((𝜑𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1413biimpd 132 . . . . . . . . 9 ((𝜑𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1514impancom 247 . . . . . . . 8 ((𝜑𝑥 = ⟨𝑎, 𝑏, 𝑐⟩) → (𝜓𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1615expimpd 345 . . . . . . 7 (𝜑 → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1716exlimdv 1700 . . . . . 6 (𝜑 → (∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1817exlimdvv 1777 . . . . 5 (𝜑 → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
19 tru 1247 . . . . . . . . . . 11
202adantr 261 . . . . . . . . . . . . 13 ((𝜑𝑎 = 𝐴) → 𝐵 ∈ V)
213ad2antrr 457 . . . . . . . . . . . . . 14 (((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝐶 ∈ V)
22 simpr 103 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
2322, 11sylibr 137 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
2423eqcomd 2045 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩)
256biimpar 281 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → 𝜓)
2624, 25jca 290 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
27 a1tru 1259 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⊤)
2826, 272thd 164 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
29283anassrs 1126 . . . . . . . . . . . . . 14 ((((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3021, 29sbcied 2799 . . . . . . . . . . . . 13 (((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3120, 30sbcied 2799 . . . . . . . . . . . 12 ((𝜑𝑎 = 𝐴) → ([𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
321, 31sbcied 2799 . . . . . . . . . . 11 (𝜑 → ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3319, 32mpbiri 157 . . . . . . . . . 10 (𝜑[𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
3433spesbcd 2844 . . . . . . . . 9 (𝜑 → ∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
35 nfcv 2178 . . . . . . . . . 10 𝑏𝐵
36 nfsbc1v 2782 . . . . . . . . . . 11 𝑏[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
3736nfex 1528 . . . . . . . . . 10 𝑏𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
38 sbceq1a 2773 . . . . . . . . . . 11 (𝑏 = 𝐵 → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
3938exbidv 1706 . . . . . . . . . 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 2636 . . . . . . . . 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 56 . . . . . . . 8 (𝜑 → ∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
42 nfcv 2178 . . . . . . . . 9 𝑐𝐶
43 nfsbc1v 2782 . . . . . . . . . . 11 𝑐[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
4443nfex 1528 . . . . . . . . . 10 𝑐𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
4544nfex 1528 . . . . . . . . 9 𝑐𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
46 sbceq1a 2773 . . . . . . . . . 10 (𝑐 = 𝐶 → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
47462exbidv 1748 . . . . . . . . 9 (𝑐 = 𝐶 → (∃𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
4842, 45, 47spcegf 2636 . . . . . . . 8 (𝐶 ∈ V → (∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → ∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
493, 41, 48sylc 56 . . . . . . 7 (𝜑 → ∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
50 excom13 1579 . . . . . . 7 (∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
5149, 50sylib 127 . . . . . 6 (𝜑 → ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
52 eqeq1 2046 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩))
5352anbi1d 438 . . . . . . 7 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
54533exbidv 1749 . . . . . 6 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
5551, 54syl5ibrcom 146 . . . . 5 (𝜑 → (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
5618, 55impbid 120 . . . 4 (𝜑 → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
5756alrimiv 1754 . . 3 (𝜑 → ∀𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
58 eqeq2 2049 . . . . . 6 (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = 𝑦𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
5958bibi2d 221 . . . . 5 (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → ((∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦) ↔ (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩)))
6059albidv 1705 . . . 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 2641 . . 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 56 . 2 (𝜑 → ∃𝑦𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
63 df-eu 1903 . 2 (∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑦𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
6462, 63sylibr 137 1 (𝜑 → ∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885  ∀wal 1241   = wceq 1243  ⊤wtru 1244  ∃wex 1381   ∈ wcel 1393  ∃!weu 1900  Vcvv 2557  [wsbc 2764  ⟨cotp 3379 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-ot 3385 This theorem is referenced by: (None)
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