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Theorem reuhypd 4399
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 2700 . . . . 5 (𝐵𝐶𝐵 ∈ V)
31, 2syl 14 . . . 4 ((𝜑𝑥𝐶) → 𝐵 ∈ V)
4 eueq 2858 . . . 4 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
53, 4sylib 121 . . 3 ((𝜑𝑥𝐶) → ∃!𝑦 𝑦 = 𝐵)
6 eleq1 2203 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝐶𝐵𝐶))
71, 6syl5ibrcom 156 . . . . . 6 ((𝜑𝑥𝐶) → (𝑦 = 𝐵𝑦𝐶))
87pm4.71rd 392 . . . . 5 ((𝜑𝑥𝐶) → (𝑦 = 𝐵 ↔ (𝑦𝐶𝑦 = 𝐵)))
9 reuhypd.2 . . . . . . 7 ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
1093expa 1182 . . . . . 6 (((𝜑𝑥𝐶) ∧ 𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
1110pm5.32da 448 . . . . 5 ((𝜑𝑥𝐶) → ((𝑦𝐶𝑥 = 𝐴) ↔ (𝑦𝐶𝑦 = 𝐵)))
128, 11bitr4d 190 . . . 4 ((𝜑𝑥𝐶) → (𝑦 = 𝐵 ↔ (𝑦𝐶𝑥 = 𝐴)))
1312eubidv 2008 . . 3 ((𝜑𝑥𝐶) → (∃!𝑦 𝑦 = 𝐵 ↔ ∃!𝑦(𝑦𝐶𝑥 = 𝐴)))
145, 13mpbid 146 . 2 ((𝜑𝑥𝐶) → ∃!𝑦(𝑦𝐶𝑥 = 𝐴))
15 df-reu 2424 . 2 (∃!𝑦𝐶 𝑥 = 𝐴 ↔ ∃!𝑦(𝑦𝐶𝑥 = 𝐴))
1614, 15sylibr 133 1 ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963   = wceq 1332  wcel 1481  ∃!weu 2000  ∃!wreu 2419  Vcvv 2689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-reu 2424  df-v 2691
This theorem is referenced by:  reuhyp  4400
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