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Theorem reu6i 2899
 Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
reu6i ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reu6i
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2164 . . . . 5 (𝑦 = 𝐵 → (𝑥 = 𝑦𝑥 = 𝐵))
21bibi2d 231 . . . 4 (𝑦 = 𝐵 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐵)))
32ralbidv 2454 . . 3 (𝑦 = 𝐵 → (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)))
43rspcev 2813 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
5 reu6 2897 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
64, 5sylibr 133 1 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 2125  ∀wral 2432  ∃wrex 2433  ∃!wreu 2434 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-reu 2439  df-v 2711 This theorem is referenced by:  eqreu  2900  riota5f  5794  negeu  8045  creur  8809  creui  8810
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