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Theorem eu1 1939
Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)
Hypothesis
Ref Expression
eu1.1 (𝜑 → ∀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 hbs1 1828 . . 3 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
21euf 1919 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
3 eu1.1 . . 3 (𝜑 → ∀𝑦𝜑)
43sb8euh 1937 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
5 equcom 1607 . . . . . . 7 (𝑥 = 𝑦𝑦 = 𝑥)
65imbi2i 219 . . . . . 6 (([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
76albii 1373 . . . . 5 (∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
83sb6rf 1747 . . . . 5 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
97, 8anbi12i 441 . . . 4 ((∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
10 ancom 257 . . . 4 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ (∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ∧ 𝜑))
11 albiim 1390 . . . 4 (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ↔ (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
129, 10, 113bitr4i 205 . . 3 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
1312exbii 1510 . 2 (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ ∃𝑥𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
142, 4, 133bitr4i 205 1 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1255  wex 1395  [wsb 1659  ∃!weu 1914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442
This theorem depends on definitions:  df-bi 114  df-nf 1364  df-sb 1660  df-eu 1917
This theorem is referenced by:  euex  1944  eu2  1958
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