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Theorem eqeu 3007
 Description: A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)
Hypothesis
Ref Expression
eqeu.1 (x = A → (φψ))
Assertion
Ref Expression
eqeu ((A B ψ x(φx = A)) → ∃!xφ)
Distinct variable groups:   ψ,x   x,A
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem eqeu
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqeu.1 . . . . 5 (x = A → (φψ))
21spcegv 2940 . . . 4 (A B → (ψxφ))
32imp 418 . . 3 ((A B ψ) → xφ)
433adant3 975 . 2 ((A B ψ x(φx = A)) → xφ)
5 eqeq2 2362 . . . . . . 7 (y = A → (x = yx = A))
65imbi2d 307 . . . . . 6 (y = A → ((φx = y) ↔ (φx = A)))
76albidv 1625 . . . . 5 (y = A → (x(φx = y) ↔ x(φx = A)))
87spcegv 2940 . . . 4 (A B → (x(φx = A) → yx(φx = y)))
98imp 418 . . 3 ((A B x(φx = A)) → yx(φx = y))
1093adant2 974 . 2 ((A B ψ x(φx = A)) → yx(φx = y))
11 nfv 1619 . . 3 yφ
1211eu3 2230 . 2 (∃!xφ ↔ (xφ yx(φx = y)))
134, 10, 12sylanbrc 645 1 ((A B ψ x(φx = A)) → ∃!xφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by: (None)
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