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Theorem reu6i 2951
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 2203 . . . . 5 (𝑦 = 𝐵 → (𝑥 = 𝑦𝑥 = 𝐵))
21bibi2d 232 . . . 4 (𝑦 = 𝐵 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐵)))
32ralbidv 2494 . . 3 (𝑦 = 𝐵 → (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)))
43rspcev 2864 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
5 reu6 2949 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
64, 5sylibr 134 1 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2164  wral 2472  wrex 2473  ∃!wreu 2474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-v 2762
This theorem is referenced by:  eqreu  2952  riota5f  5898  negeu  8210  creur  8978  creui  8979
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