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Theorem euotd 4940
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.4 . . . . . . . . . . . . 13 (𝜑 → (𝜓 ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)))
21biimpa 501 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
3 vex 3192 . . . . . . . . . . . . 13 𝑎 ∈ V
4 vex 3192 . . . . . . . . . . . . 13 𝑏 ∈ V
5 vex 3192 . . . . . . . . . . . . 13 𝑐 ∈ V
63, 4, 5otth 4918 . . . . . . . . . . . 12 (⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩ ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
72, 6sylibr 224 . . . . . . . . . . 11 ((𝜑𝜓) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
87eqeq2d 2631 . . . . . . . . . 10 ((𝜑𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
98biimpd 219 . . . . . . . . 9 ((𝜑𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
109impancom 456 . . . . . . . 8 ((𝜑𝑥 = ⟨𝑎, 𝑏, 𝑐⟩) → (𝜓𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1110expimpd 628 . . . . . . 7 (𝜑 → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1211exlimdv 1858 . . . . . 6 (𝜑 → (∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1312exlimdvv 1859 . . . . 5 (𝜑 → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
14 euotd.3 . . . . . . . 8 (𝜑𝐶 ∈ V)
15 euotd.2 . . . . . . . . 9 (𝜑𝐵 ∈ V)
16 tru 1484 . . . . . . . . . . 11
17 euotd.1 . . . . . . . . . . . 12 (𝜑𝐴 ∈ V)
1815adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑎 = 𝐴) → 𝐵 ∈ V)
1914ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝐶 ∈ V)
20 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
2120, 6sylibr 224 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
2221eqcomd 2627 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩)
231biimpar 502 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → 𝜓)
2422, 23jca 554 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
25 a1tru 1497 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⊤)
2624, 252thd 255 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
27263anassrs 1287 . . . . . . . . . . . . . 14 ((((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
2819, 27sbcied 3458 . . . . . . . . . . . . 13 (((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
2918, 28sbcied 3458 . . . . . . . . . . . 12 ((𝜑𝑎 = 𝐴) → ([𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3017, 29sbcied 3458 . . . . . . . . . . 11 (𝜑 → ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3116, 30mpbiri 248 . . . . . . . . . 10 (𝜑[𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
3231spesbcd 3507 . . . . . . . . 9 (𝜑 → ∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
33 nfcv 2761 . . . . . . . . . 10 𝑏𝐵
34 nfsbc1v 3441 . . . . . . . . . . 11 𝑏[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
3534nfex 2151 . . . . . . . . . 10 𝑏𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
36 sbceq1a 3432 . . . . . . . . . . 11 (𝑏 = 𝐵 → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
3736exbidv 1847 . . . . . . . . . 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)))
3833, 35, 37spcegf 3278 . . . . . . . . 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)))
3915, 32, 38sylc 65 . . . . . . . 8 (𝜑 → ∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
40 nfcv 2761 . . . . . . . . 9 𝑐𝐶
41 nfsbc1v 3441 . . . . . . . . . . 11 𝑐[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
4241nfex 2151 . . . . . . . . . 10 𝑐𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
4342nfex 2151 . . . . . . . . 9 𝑐𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
44 sbceq1a 3432 . . . . . . . . . 10 (𝑐 = 𝐶 → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
45442exbidv 1849 . . . . . . . . 9 (𝑐 = 𝐶 → (∃𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
4640, 43, 45spcegf 3278 . . . . . . . 8 (𝐶 ∈ V → (∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → ∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
4714, 39, 46sylc 65 . . . . . . 7 (𝜑 → ∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
48 excom13 2041 . . . . . . 7 (∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
4947, 48sylib 208 . . . . . 6 (𝜑 → ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
50 eqeq1 2625 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩))
5150anbi1d 740 . . . . . . 7 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
52513exbidv 1850 . . . . . 6 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
5349, 52syl5ibrcom 237 . . . . 5 (𝜑 → (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
5413, 53impbid 202 . . . 4 (𝜑 → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
5554alrimiv 1852 . . 3 (𝜑 → ∀𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
56 otex 4899 . . . 4 𝐴, 𝐵, 𝐶⟩ ∈ V
57 eqeq2 2632 . . . . . 6 (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = 𝑦𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
5857bibi2d 332 . . . . 5 (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → ((∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦) ↔ (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩)))
5958albidv 1846 . . . 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⟩)))
6056, 59spcev 3289 . . 3 (∀𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩) → ∃𝑦𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
6155, 60syl 17 . 2 (𝜑 → ∃𝑦𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
62 df-eu 2473 . 2 (∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑦𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
6361, 62sylibr 224 1 (𝜑 → ∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wtru 1481  wex 1701  wcel 1987  ∃!weu 2469  Vcvv 3189  [wsbc 3421  cotp 4161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-ot 4162
This theorem is referenced by:  oeeu  7635
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