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Theorem reuhypd 4230
Description: A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.)
Hypotheses
Ref Expression
reuhypd.1 ((𝜑𝑥𝐶) → 𝐵𝐶)
reuhypd.2 ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
Assertion
Ref Expression
reuhypd ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
Distinct variable groups:   𝜑,𝑦   𝑦,𝐵   𝑦,𝐶   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem reuhypd
StepHypRef Expression
1 reuhypd.1 . . . . 5 ((𝜑𝑥𝐶) → 𝐵𝐶)
2 elex 2583 . . . . 5 (𝐵𝐶𝐵 ∈ V)
31, 2syl 14 . . . 4 ((𝜑𝑥𝐶) → 𝐵 ∈ V)
4 eueq 2734 . . . 4 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
53, 4sylib 131 . . 3 ((𝜑𝑥𝐶) → ∃!𝑦 𝑦 = 𝐵)
6 eleq1 2116 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝐶𝐵𝐶))
71, 6syl5ibrcom 150 . . . . . 6 ((𝜑𝑥𝐶) → (𝑦 = 𝐵𝑦𝐶))
87pm4.71rd 380 . . . . 5 ((𝜑𝑥𝐶) → (𝑦 = 𝐵 ↔ (𝑦𝐶𝑦 = 𝐵)))
9 reuhypd.2 . . . . . . 7 ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
1093expa 1115 . . . . . 6 (((𝜑𝑥𝐶) ∧ 𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
1110pm5.32da 433 . . . . 5 ((𝜑𝑥𝐶) → ((𝑦𝐶𝑥 = 𝐴) ↔ (𝑦𝐶𝑦 = 𝐵)))
128, 11bitr4d 184 . . . 4 ((𝜑𝑥𝐶) → (𝑦 = 𝐵 ↔ (𝑦𝐶𝑥 = 𝐴)))
1312eubidv 1924 . . 3 ((𝜑𝑥𝐶) → (∃!𝑦 𝑦 = 𝐵 ↔ ∃!𝑦(𝑦𝐶𝑥 = 𝐴)))
145, 13mpbid 139 . 2 ((𝜑𝑥𝐶) → ∃!𝑦(𝑦𝐶𝑥 = 𝐴))
15 df-reu 2330 . 2 (∃!𝑦𝐶 𝑥 = 𝐴 ↔ ∃!𝑦(𝑦𝐶𝑥 = 𝐴))
1614, 15sylibr 141 1 ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  ∃!weu 1916  ∃!wreu 2325  Vcvv 2574
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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-reu 2330  df-v 2576
This theorem is referenced by:  reuhyp  4231
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