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Theorem eu1 1973
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 1862 . . 3 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
21euf 1953 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
3 eu1.1 . . 3 (𝜑 → ∀𝑦𝜑)
43sb8euh 1971 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
5 equcom 1639 . . . . . . 7 (𝑥 = 𝑦𝑦 = 𝑥)
65imbi2i 224 . . . . . 6 (([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
76albii 1404 . . . . 5 (∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
83sb6rf 1781 . . . . 5 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
97, 8anbi12i 448 . . . 4 ((∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
10 ancom 262 . . . 4 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ (∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ∧ 𝜑))
11 albiim 1421 . . . 4 (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ↔ (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
129, 10, 113bitr4i 210 . . 3 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
1312exbii 1541 . 2 (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ ∃𝑥𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
142, 4, 133bitr4i 210 1 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287  wex 1426  [wsb 1692  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951
This theorem is referenced by:  euex  1978  eu2  1992
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