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Theorem eqeu 2711
Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
Hypothesis
Ref Expression
eqeu.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
eqeu ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑)
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem eqeu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeu.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
21spcegv 2641 . . . 4 (𝐴𝐵 → (𝜓 → ∃𝑥𝜑))
32imp 115 . . 3 ((𝐴𝐵𝜓) → ∃𝑥𝜑)
433adant3 924 . 2 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑥𝜑)
5 eqeq2 2049 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
65imbi2d 219 . . . . . 6 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
76albidv 1705 . . . . 5 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
87spcegv 2641 . . . 4 (𝐴𝐵 → (∀𝑥(𝜑𝑥 = 𝐴) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
98imp 115 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
1093adant2 923 . 2 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
11 nfv 1421 . . 3 𝑦𝜑
1211eu3 1946 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
134, 10, 12sylanbrc 394 1 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  w3a 885  wal 1241   = wceq 1243  wex 1381  wcel 1393  ∃!weu 1900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by: (None)
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