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Theorem reuhypd 5425
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 7422. (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)
21elexd 3502 . . . 4 ((𝜑𝑥𝐶) → 𝐵 ∈ V)
3 eueq 3717 . . . 4 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
42, 3sylib 218 . . 3 ((𝜑𝑥𝐶) → ∃!𝑦 𝑦 = 𝐵)
5 eleq1 2827 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝐶𝐵𝐶))
61, 5syl5ibrcom 247 . . . . . 6 ((𝜑𝑥𝐶) → (𝑦 = 𝐵𝑦𝐶))
76pm4.71rd 562 . . . . 5 ((𝜑𝑥𝐶) → (𝑦 = 𝐵 ↔ (𝑦𝐶𝑦 = 𝐵)))
8 reuhypd.2 . . . . . . 7 ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
983expa 1117 . . . . . 6 (((𝜑𝑥𝐶) ∧ 𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
109pm5.32da 579 . . . . 5 ((𝜑𝑥𝐶) → ((𝑦𝐶𝑥 = 𝐴) ↔ (𝑦𝐶𝑦 = 𝐵)))
117, 10bitr4d 282 . . . 4 ((𝜑𝑥𝐶) → (𝑦 = 𝐵 ↔ (𝑦𝐶𝑥 = 𝐴)))
1211eubidv 2584 . . 3 ((𝜑𝑥𝐶) → (∃!𝑦 𝑦 = 𝐵 ↔ ∃!𝑦(𝑦𝐶𝑥 = 𝐴)))
134, 12mpbid 232 . 2 ((𝜑𝑥𝐶) → ∃!𝑦(𝑦𝐶𝑥 = 𝐴))
14 df-reu 3379 . 2 (∃!𝑦𝐶 𝑥 = 𝐴 ↔ ∃!𝑦(𝑦𝐶𝑥 = 𝐴))
1513, 14sylibr 234 1 ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  ∃!weu 2566  ∃!wreu 3376  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-reu 3379  df-v 3480
This theorem is referenced by:  reuhyp  5426  riotaocN  39191
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