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Theorem eu3h 1961
Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)
Hypothesis
Ref Expression
eu3h.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
eu3h (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu3h
StepHypRef Expression
1 euex 1946 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
2 eu3h.1 . . . 4 (𝜑 → ∀𝑦𝜑)
32eumo0 1947 . . 3 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
41, 3jca 294 . 2 (∃!𝑥𝜑 → (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
52nfi 1367 . . . . 5 𝑦𝜑
65mo23 1957 . . . 4 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
76anim2i 328 . . 3 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
85eu2 1960 . . 3 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
97, 8sylibr 141 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ∃!𝑥𝜑)
104, 9impbii 121 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  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:  eu3  1962  mo2r  1968  2eu4  2009
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