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Theorem reu6i 3384
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 2637 . . . . 5 (𝑦 = 𝐵 → (𝑥 = 𝑦𝑥 = 𝐵))
21bibi2d 332 . . . 4 (𝑦 = 𝐵 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐵)))
32ralbidv 2985 . . 3 (𝑦 = 𝐵 → (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)))
43rspcev 3300 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
5 reu6 3382 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
64, 5sylibr 224 1 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  ∃!wreu 2914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-reu 2919  df-v 3193
This theorem is referenced by:  eqreu  3385  riota5f  6591  negeu  10216  creur  10959  creui  10960  reuccats1  13413  lublecl  16905  dfod2  17897  lmieu  25571  esum2dlem  29927  poimirlem16  33043  poimirlem17  33044  poimirlem19  33046  poimirlem20  33047  poimirlem22  33049  reuccatpfxs1  40721
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