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Theorem reu3 2845
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 2619 . . 3 (∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)
2 reu6 2844 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
3 bi1 117 . . . . . 6 ((𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
43ralimi 2470 . . . . 5 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) → ∀𝑥𝐴 (𝜑𝑥 = 𝑦))
54reximi 2504 . . . 4 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
62, 5sylbi 120 . . 3 (∃!𝑥𝐴 𝜑 → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
71, 6jca 302 . 2 (∃!𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
8 rexex 2454 . . . 4 (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))
98anim2i 337 . . 3 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)) → (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)))
10 nfv 1491 . . . . 5 𝑦(𝑥𝐴𝜑)
1110eu3 2021 . . . 4 (∃!𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦)))
12 df-reu 2398 . . . 4 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
13 df-rex 2397 . . . . 5 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
14 df-ral 2396 . . . . . . 7 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
15 impexp 261 . . . . . . . 8 (((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝑦)))
1615albii 1429 . . . . . . 7 (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥 = 𝑦)))
1714, 16bitr4i 186 . . . . . 6 (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
1817exbii 1567 . . . . 5 (∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦))
1913, 18anbi12i 453 . . . 4 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑦)))
2011, 12, 193bitr4i 211 . . 3 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦)))
219, 20sylibr 133 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)) → ∃!𝑥𝐴 𝜑)
227, 21impbii 125 1 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312  wex 1451  wcel 1463  ∃!weu 1975  wral 2391  wrex 2392  ∃!wreu 2393
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-cleq 2108  df-clel 2111  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399
This theorem is referenced by:  reu7  2850  bdreu  12887
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