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Theorem eu4 2613
Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
Hypothesis
Ref Expression
eu4.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
eu4 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem eu4
StepHypRef Expression
1 df-eu 2567 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2 eu4.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32mo4 2564 . . 3 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
43anbi2i 623 . 2 ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦)))
51, 4bitri 275 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wex 1776  ∃*wmo 2536  ∃!weu 2566
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-mo 2538  df-eu 2567
This theorem is referenced by:  euind  3733  eqeuel  4371  uniintsn  4990  eusv1  5397  omeu  8622  eroveu  8851  climeu  15588  pceu  16880  initoeu2lem2  18069  psgneu  19539  gsumval3eu  19937  frgr3vlem2  30303  3vfriswmgrlem  30306  unirep  37701  rlimdmafv  47127  rlimdmafv2  47208
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