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Theorem eqeu 2849
 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 2769 . . . 4 (𝐴𝐵 → (𝜓 → ∃𝑥𝜑))
32imp 123 . . 3 ((𝐴𝐵𝜓) → ∃𝑥𝜑)
433adant3 1001 . 2 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑥𝜑)
5 eqeq2 2147 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
65imbi2d 229 . . . . . 6 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
76albidv 1796 . . . . 5 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
87spcegv 2769 . . . 4 (𝐴𝐵 → (∀𝑥(𝜑𝑥 = 𝐴) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
98imp 123 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
1093adant2 1000 . 2 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
11 nfv 1508 . . 3 𝑦𝜑
1211eu3 2043 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
134, 10, 12sylanbrc 413 1 ((𝐴𝐵𝜓 ∧ ∀𝑥(𝜑𝑥 = 𝐴)) → ∃!𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 962  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃!weu 1997 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683 This theorem is referenced by: (None)
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