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Theorem euotd 4018
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 3994 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
51, 2, 3, 4syl3anc 1146 . . 3 (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
6 euotd.4 . . . . . . . . . . . . 13 (𝜑 → (𝜓 ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)))
76biimpa 284 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
8 vex 2577 . . . . . . . . . . . . 13 𝑎 ∈ V
9 vex 2577 . . . . . . . . . . . . 13 𝑏 ∈ V
10 vex 2577 . . . . . . . . . . . . 13 𝑐 ∈ V
118, 9, 10otth 4006 . . . . . . . . . . . 12 (⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩ ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
127, 11sylibr 141 . . . . . . . . . . 11 ((𝜑𝜓) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
1312eqeq2d 2067 . . . . . . . . . 10 ((𝜑𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1413biimpd 136 . . . . . . . . 9 ((𝜑𝜓) → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1514impancom 251 . . . . . . . 8 ((𝜑𝑥 = ⟨𝑎, 𝑏, 𝑐⟩) → (𝜓𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1615expimpd 349 . . . . . . 7 (𝜑 → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1716exlimdv 1716 . . . . . 6 (𝜑 → (∃𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
1817exlimdvv 1793 . . . . 5 (𝜑 → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
19 tru 1263 . . . . . . . . . . 11
202adantr 265 . . . . . . . . . . . . 13 ((𝜑𝑎 = 𝐴) → 𝐵 ∈ V)
213ad2antrr 465 . . . . . . . . . . . . . 14 (((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝐶 ∈ V)
22 simpr 107 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶))
2322, 11sylibr 141 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
2423eqcomd 2061 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩)
256biimpar 285 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → 𝜓)
2624, 25jca 294 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
27 a1tru 1275 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ⊤)
2826, 272thd 168 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
29283anassrs 1137 . . . . . . . . . . . . . 14 ((((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3021, 29sbcied 2821 . . . . . . . . . . . . 13 (((𝜑𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3120, 30sbcied 2821 . . . . . . . . . . . 12 ((𝜑𝑎 = 𝐴) → ([𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
321, 31sbcied 2821 . . . . . . . . . . 11 (𝜑 → ([𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ⊤))
3319, 32mpbiri 161 . . . . . . . . . 10 (𝜑[𝐴 / 𝑎][𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
3433spesbcd 2871 . . . . . . . . 9 (𝜑 → ∃𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
35 nfcv 2194 . . . . . . . . . 10 𝑏𝐵
36 nfsbc1v 2804 . . . . . . . . . . 11 𝑏[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
3736nfex 1544 . . . . . . . . . 10 𝑏𝑎[𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
38 sbceq1a 2795 . . . . . . . . . . 11 (𝑏 = 𝐵 → ([𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐵 / 𝑏][𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
3938exbidv 1722 . . . . . . . . . 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 2653 . . . . . . . . 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 60 . . . . . . . 8 (𝜑 → ∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
42 nfcv 2194 . . . . . . . . 9 𝑐𝐶
43 nfsbc1v 2804 . . . . . . . . . . 11 𝑐[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
4443nfex 1544 . . . . . . . . . 10 𝑐𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
4544nfex 1544 . . . . . . . . 9 𝑐𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)
46 sbceq1a 2795 . . . . . . . . . 10 (𝑐 = 𝐶 → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ [𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
47462exbidv 1764 . . . . . . . . 9 (𝑐 = 𝐶 → (∃𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
4842, 45, 47spcegf 2653 . . . . . . . 8 (𝐶 ∈ V → (∃𝑏𝑎[𝐶 / 𝑐](⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) → ∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
493, 41, 48sylc 60 . . . . . . 7 (𝜑 → ∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
50 excom13 1595 . . . . . . 7 (∃𝑐𝑏𝑎(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
5149, 50sylib 131 . . . . . 6 (𝜑 → ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
52 eqeq1 2062 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ↔ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩))
5352anbi1d 446 . . . . . . 7 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
54533exbidv 1765 . . . . . 6 (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑎𝑏𝑐(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
5551, 54syl5ibrcom 150 . . . . 5 (𝜑 → (𝑥 = ⟨𝐴, 𝐵, 𝐶⟩ → ∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓)))
5618, 55impbid 124 . . . 4 (𝜑 → (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
5756alrimiv 1770 . . 3 (𝜑 → ∀𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
58 eqeq2 2065 . . . . . 6 (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑥 = 𝑦𝑥 = ⟨𝐴, 𝐵, 𝐶⟩))
5958bibi2d 225 . . . . 5 (𝑦 = ⟨𝐴, 𝐵, 𝐶⟩ → ((∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦) ↔ (∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = ⟨𝐴, 𝐵, 𝐶⟩)))
6059albidv 1721 . . . 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 2658 . . 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 60 . 2 (𝜑 → ∃𝑦𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
63 df-eu 1919 . 2 (∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ ∃𝑦𝑥(∃𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓) ↔ 𝑥 = 𝑦))
6462, 63sylibr 141 1 (𝜑 → ∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896  wal 1257   = wceq 1259  wtru 1260  wex 1397  wcel 1409  ∃!weu 1916  Vcvv 2574  [wsbc 2786  cotp 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-ot 3412
This theorem is referenced by: (None)
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