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Theorem reu3 2751
Description: A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)
Assertion
Ref Expression
reu3 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reu3
StepHypRef Expression
1 reurex 2538 . . 3 (∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)
2 reu6 2750 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
3 bi1 115 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
43ralimi 2399 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) → ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
54reximi 2431 . . . 4 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
62, 5sylbi 118 . . 3 (∃!𝑥𝐴 𝜑 → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
71, 6jca 294 . 2 (∃!𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
8 rexex 2383 . . . 4 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
98anim2i 328 . . 3 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)) → (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)))
10 nfv 1435 . . . . 5 𝑦(𝑥𝐴𝜑)
1110eu3 1960 . . . 4 (∃!𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦)))
12 df-reu 2328 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
13 df-rex 2327 . . . . 5 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
14 df-ral 2326 . . . . . . 7 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
15 impexp 254 . . . . . . . 8 (((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
1615albii 1373 . . . . . . 7 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
1714, 16bitr4i 180 . . . . . 6 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
1817exbii 1510 . . . . 5 (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
1913, 18anbi12i 441 . . . 4 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦)))
2011, 12, 193bitr4i 205 . . 3 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)))
219, 20sylibr 141 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)) → ∃!𝑥𝐴 𝜑)
227, 21impbii 121 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1255  wex 1395  wcel 1407  ∃!weu 1914  wral 2321  wrex 2322  ∃!wreu 2323
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  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-cleq 2047  df-clel 2050  df-ral 2326  df-rex 2327  df-reu 2328  df-rmo 2329
This theorem is referenced by:  reu7  2756  bdreu  10305
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