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Theorem reu6i 2929
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 2187 . . . . 5 (𝑦 = 𝐵 → (𝑥 = 𝑦𝑥 = 𝐵))
21bibi2d 232 . . . 4 (𝑦 = 𝐵 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐵)))
32ralbidv 2477 . . 3 (𝑦 = 𝐵 → (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)))
43rspcev 2842 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
5 reu6 2927 . 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 1353  wcel 2148  wral 2455  wrex 2456  ∃!wreu 2457
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2740
This theorem is referenced by:  eqreu  2930  riota5f  5855  negeu  8148  creur  8916  creui  8917
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