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Theorem reu6i 2726
 Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
reu6i ((B A x A (φx = B)) → ∃!x A 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 2046 . . . . 5 (y = B → (x = yx = B))
21bibi2d 221 . . . 4 (y = B → ((φx = y) ↔ (φx = B)))
32ralbidv 2320 . . 3 (y = B → (x A (φx = y) ↔ x A (φx = B)))
43rspcev 2650 . 2 ((B A x A (φx = B)) → y A x A (φx = y))
5 reu6 2724 . 2 (∃!x A φy A x A (φx = y))
64, 5sylibr 137 1 ((B A x A (φx = B)) → ∃!x A φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  ∃!wreu 2302 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553 This theorem is referenced by:  eqreu  2727  riota5f  5435  negeu  6999  creur  7692  creui  7693
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