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Theorem euotd 3961
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 = A 𝑏 = B 𝑐 = 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 3937 . . . 4 ((A V B V 𝐶 V) → ⟨A, B, 𝐶 V)
51, 2, 3, 4syl3anc 1119 . . 3 (φ → ⟨A, B, 𝐶 V)
6 euotd.4 . . . . . . . . . . . . 13 (φ → (ψ ↔ (𝑎 = A 𝑏 = B 𝑐 = 𝐶)))
76biimpa 280 . . . . . . . . . . . 12 ((φ ψ) → (𝑎 = A 𝑏 = B 𝑐 = 𝐶))
8 vex 2534 . . . . . . . . . . . . 13 𝑎 V
9 vex 2534 . . . . . . . . . . . . 13 𝑏 V
10 vex 2534 . . . . . . . . . . . . 13 𝑐 V
118, 9, 10otth 3949 . . . . . . . . . . . 12 (⟨𝑎, 𝑏, 𝑐⟩ = ⟨A, B, 𝐶⟩ ↔ (𝑎 = A 𝑏 = B 𝑐 = 𝐶))
127, 11sylibr 137 . . . . . . . . . . 11 ((φ ψ) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨A, B, 𝐶⟩)
1312eqeq2d 2029 . . . . . . . . . 10 ((φ ψ) → (x = ⟨𝑎, 𝑏, 𝑐⟩ ↔ x = ⟨A, B, 𝐶⟩))
1413biimpd 132 . . . . . . . . 9 ((φ ψ) → (x = ⟨𝑎, 𝑏, 𝑐⟩ → x = ⟨A, B, 𝐶⟩))
1514impancom 247 . . . . . . . 8 ((φ x = ⟨𝑎, 𝑏, 𝑐⟩) → (ψx = ⟨A, B, 𝐶⟩))
1615expimpd 345 . . . . . . 7 (φ → ((x = ⟨𝑎, 𝑏, 𝑐 ψ) → x = ⟨A, B, 𝐶⟩))
1716exlimdv 1678 . . . . . 6 (φ → (𝑐(x = ⟨𝑎, 𝑏, 𝑐 ψ) → x = ⟨A, B, 𝐶⟩))
1817exlimdvv 1755 . . . . 5 (φ → (𝑎𝑏𝑐(x = ⟨𝑎, 𝑏, 𝑐 ψ) → x = ⟨A, B, 𝐶⟩))
19 tru 1230 . . . . . . . . . . 11
202adantr 261 . . . . . . . . . . . . 13 ((φ 𝑎 = A) → B V)
213ad2antrr 460 . . . . . . . . . . . . . 14 (((φ 𝑎 = A) 𝑏 = B) → 𝐶 V)
22 simpr 103 . . . . . . . . . . . . . . . . . . 19 ((φ (𝑎 = A 𝑏 = B 𝑐 = 𝐶)) → (𝑎 = A 𝑏 = B 𝑐 = 𝐶))
2322, 11sylibr 137 . . . . . . . . . . . . . . . . . 18 ((φ (𝑎 = A 𝑏 = B 𝑐 = 𝐶)) → ⟨𝑎, 𝑏, 𝑐⟩ = ⟨A, B, 𝐶⟩)
2423eqcomd 2023 . . . . . . . . . . . . . . . . 17 ((φ (𝑎 = A 𝑏 = B 𝑐 = 𝐶)) → ⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩)
256biimpar 281 . . . . . . . . . . . . . . . . 17 ((φ (𝑎 = A 𝑏 = B 𝑐 = 𝐶)) → ψ)
2624, 25jca 290 . . . . . . . . . . . . . . . 16 ((φ (𝑎 = A 𝑏 = B 𝑐 = 𝐶)) → (⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ))
27 a1tru 1242 . . . . . . . . . . . . . . . 16 ((φ (𝑎 = A 𝑏 = B 𝑐 = 𝐶)) → ⊤ )
2826, 272thd 164 . . . . . . . . . . . . . . 15 ((φ (𝑎 = A 𝑏 = B 𝑐 = 𝐶)) → ((⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ ⊤ ))
29283anassrs 1110 . . . . . . . . . . . . . 14 ((((φ 𝑎 = A) 𝑏 = B) 𝑐 = 𝐶) → ((⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ ⊤ ))
3021, 29sbcied 2772 . . . . . . . . . . . . 13 (((φ 𝑎 = A) 𝑏 = B) → ([𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ ⊤ ))
3120, 30sbcied 2772 . . . . . . . . . . . 12 ((φ 𝑎 = A) → ([B / 𝑏][𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ ⊤ ))
321, 31sbcied 2772 . . . . . . . . . . 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 2817 . . . . . . . . 9 (φ𝑎[B / 𝑏][𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ))
35 nfcv 2156 . . . . . . . . . 10 𝑏B
36 nfsbc1v 2755 . . . . . . . . . . 11 𝑏[B / 𝑏][𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ)
3736nfex 1506 . . . . . . . . . 10 𝑏𝑎[B / 𝑏][𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ)
38 sbceq1a 2746 . . . . . . . . . . 11 (𝑏 = B → ([𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ [B / 𝑏][𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ)))
3938exbidv 1684 . . . . . . . . . 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 2609 . . . . . . . . 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 (φ𝑏𝑎[𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ))
42 nfcv 2156 . . . . . . . . 9 𝑐𝐶
43 nfsbc1v 2755 . . . . . . . . . . 11 𝑐[𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ)
4443nfex 1506 . . . . . . . . . 10 𝑐𝑎[𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ)
4544nfex 1506 . . . . . . . . 9 𝑐𝑏𝑎[𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ)
46 sbceq1a 2746 . . . . . . . . . 10 (𝑐 = 𝐶 → ((⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ [𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ)))
47462exbidv 1726 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑏𝑎(⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ 𝑏𝑎[𝐶 / 𝑐](⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ)))
4842, 45, 47spcegf 2609 . . . . . . . 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 (φ𝑐𝑏𝑎(⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ))
50 excom13 1557 . . . . . . 7 (𝑐𝑏𝑎(⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ 𝑎𝑏𝑐(⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ))
5149, 50sylib 127 . . . . . 6 (φ𝑎𝑏𝑐(⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ))
52 eqeq1 2024 . . . . . . . 8 (x = ⟨A, B, 𝐶⟩ → (x = ⟨𝑎, 𝑏, 𝑐⟩ ↔ ⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐⟩))
5352anbi1d 441 . . . . . . 7 (x = ⟨A, B, 𝐶⟩ → ((x = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ (⟨A, B, 𝐶⟩ = ⟨𝑎, 𝑏, 𝑐 ψ)))
54533exbidv 1727 . . . . . 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 1732 . . 3 (φx(𝑎𝑏𝑐(x = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ x = ⟨A, B, 𝐶⟩))
58 eqeq2 2027 . . . . . 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 1683 . . . 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 2614 . . 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 (φyx(𝑎𝑏𝑐(x = ⟨𝑎, 𝑏, 𝑐 ψ) ↔ x = y))
63 df-eu 1881 . 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 871  wal 1224   = wceq 1226  wtru 1227  wex 1358   wcel 1370  ∃!weu 1878  Vcvv 2531  [wsbc 2737  cotp 3350
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-ot 3356
This theorem is referenced by: (None)
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