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Theorem eu2 1960
 Description: An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eu2.1 𝑦𝜑
Assertion
Ref Expression
eu2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu2
StepHypRef Expression
1 euex 1946 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
2 eu2.1 . . . . . 6 𝑦𝜑
32nfri 1428 . . . . 5 (𝜑 → ∀𝑦𝜑)
43eumo0 1947 . . . 4 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
52mo23 1957 . . . 4 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
64, 5syl 14 . . 3 (∃!𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
71, 6jca 294 . 2 (∃!𝑥𝜑 → (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8 19.29r 1528 . . . 4 ((∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃𝑥(𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
9 impexp 254 . . . . . . . . 9 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
109albii 1375 . . . . . . . 8 (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦(𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
11219.21 1491 . . . . . . . 8 (∀𝑦(𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
1210, 11bitri 177 . . . . . . 7 (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
1312anbi2i 438 . . . . . 6 ((𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝜑 ∧ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦))))
14 abai 502 . . . . . 6 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ (𝜑 ∧ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦))))
1513, 14bitr4i 180 . . . . 5 ((𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
1615exbii 1512 . . . 4 (∃𝑥(𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
178, 16sylib 131 . . 3 ((∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
183eu1 1941 . . 3 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
1917, 18sylibr 141 . 2 ((∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃!𝑥𝜑)
207, 19impbii 121 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102  ∀wal 1257  Ⅎwnf 1365  ∃wex 1397  [wsb 1661  ∃!weu 1916 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 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919 This theorem is referenced by:  eu3h  1961  mo3h  1969  bm1.1  2041  reu2  2751
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