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Theorem eqeu 2785
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 2707 . . . 4 (𝐴𝐵 → (𝜓 → ∃𝑥𝜑))
32imp 122 . . 3 ((𝐴𝐵𝜓) → ∃𝑥𝜑)
433adant3 963 . 2 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑥𝜑)
5 eqeq2 2097 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
65imbi2d 228 . . . . . 6 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
76albidv 1752 . . . . 5 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
87spcegv 2707 . . . 4 (𝐴𝐵 → (∀𝑥(𝜑𝑥 = 𝐴) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
98imp 122 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
1093adant2 962 . 2 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
11 nfv 1466 . . 3 𝑦𝜑
1211eu3 1994 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
134, 10, 12sylanbrc 408 1 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  w3a 924  wal 1287   = wceq 1289  wex 1426  wcel 1438  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by: (None)
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