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Theorem reuhypd 5323
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 7151. (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 3517 . . . 4 ((𝜑𝑥𝐶) → 𝐵 ∈ V)
3 eueq 3702 . . . 4 (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵)
42, 3sylib 220 . . 3 ((𝜑𝑥𝐶) → ∃!𝑦 𝑦 = 𝐵)
5 eleq1 2903 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝐶𝐵𝐶))
61, 5syl5ibrcom 249 . . . . . 6 ((𝜑𝑥𝐶) → (𝑦 = 𝐵𝑦𝐶))
76pm4.71rd 565 . . . . 5 ((𝜑𝑥𝐶) → (𝑦 = 𝐵 ↔ (𝑦𝐶𝑦 = 𝐵)))
8 reuhypd.2 . . . . . . 7 ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
983expa 1114 . . . . . 6 (((𝜑𝑥𝐶) ∧ 𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
109pm5.32da 581 . . . . 5 ((𝜑𝑥𝐶) → ((𝑦𝐶𝑥 = 𝐴) ↔ (𝑦𝐶𝑦 = 𝐵)))
117, 10bitr4d 284 . . . 4 ((𝜑𝑥𝐶) → (𝑦 = 𝐵 ↔ (𝑦𝐶𝑥 = 𝐴)))
1211eubidv 2671 . . 3 ((𝜑𝑥𝐶) → (∃!𝑦 𝑦 = 𝐵 ↔ ∃!𝑦(𝑦𝐶𝑥 = 𝐴)))
134, 12mpbid 234 . 2 ((𝜑𝑥𝐶) → ∃!𝑦(𝑦𝐶𝑥 = 𝐴))
14 df-reu 3148 . 2 (∃!𝑦𝐶 𝑥 = 𝐴 ↔ ∃!𝑦(𝑦𝐶𝑥 = 𝐴))
1513, 14sylibr 236 1 ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1536  wcel 2113  ∃!weu 2652  ∃!wreu 3143  Vcvv 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2796
This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085  df-ex 1780  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-reu 3148  df-v 3499
This theorem is referenced by:  reuhyp  5324  riotaocN  36349
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