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Theorem eu2 2044
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 2030 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
2 eu2.1 . . . . . 6 𝑦𝜑
32nfri 1500 . . . . 5 (𝜑 → ∀𝑦𝜑)
43eumo0 2031 . . . 4 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
52mo23 2041 . . . 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 304 . 2 (∃!𝑥𝜑 → (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8 19.29r 1601 . . . 4 ((∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃𝑥(𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
9 impexp 261 . . . . . . . . 9 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
109albii 1447 . . . . . . . 8 (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑦(𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
11219.21 1563 . . . . . . . 8 (∀𝑦(𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
1210, 11bitri 183 . . . . . . 7 (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
1312anbi2i 453 . . . . . 6 ((𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝜑 ∧ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦))))
14 abai 550 . . . . . 6 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ (𝜑 ∧ (𝜑 → ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦))))
1513, 14bitr4i 186 . . . . 5 ((𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
1615exbii 1585 . . . 4 (∃𝑥(𝜑 ∧ ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
178, 16sylib 121 . . 3 ((∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
183eu1 2025 . . 3 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
1917, 18sylibr 133 . 2 ((∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∃!𝑥𝜑)
207, 19impbii 125 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1330  wnf 1437  wex 1469  [wsb 1736  ∃!weu 2000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003
This theorem is referenced by:  eu3h  2045  mo3h  2053  bm1.1  2125  reu2  2875
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