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Theorem eu1 2498
 Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.)
Hypothesis
Ref Expression
eu1.1 𝑦𝜑
Assertion
Ref Expression
eu1 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu1
StepHypRef Expression
1 nfs1v 2425 . . 3 𝑥[𝑦 / 𝑥]𝜑
21euf 2466 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
3 eu1.1 . . 3 𝑦𝜑
43sb8eu 2491 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
53sb6rf 2411 . . . . 5 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
6 equcom 1932 . . . . . . 7 (𝑥 = 𝑦𝑦 = 𝑥)
76imbi2i 325 . . . . . 6 (([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
87albii 1737 . . . . 5 (∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
95, 8anbi12ci 730 . . . 4 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
10 albiim 1806 . . . 4 (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ↔ (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
119, 10bitr4i 266 . . 3 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
1211exbii 1764 . 2 (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ ∃𝑥𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
132, 4, 123bitr4i 291 1 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699  [wsb 1867  ∃!weu 2458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462 This theorem is referenced by:  euexALT  2499  kmlem15  8869
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