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Theorem euf 2210
 Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)
Hypothesis
Ref Expression
euf.1 yφ
Assertion
Ref Expression
euf (∃!xφyx(φx = y))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem euf
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2208 . 2 (∃!xφzx(φx = z))
2 euf.1 . . . . 5 yφ
3 nfv 1619 . . . . 5 y x = z
42, 3nfbi 1834 . . . 4 y(φx = z)
54nfal 1842 . . 3 yx(φx = z)
6 nfv 1619 . . . . 5 zφ
7 nfv 1619 . . . . 5 z x = y
86, 7nfbi 1834 . . . 4 z(φx = y)
98nfal 1842 . . 3 zx(φx = y)
10 equequ2 1686 . . . . 5 (z = y → (x = zx = y))
1110bibi2d 309 . . . 4 (z = y → ((φx = z) ↔ (φx = y)))
1211albidv 1625 . . 3 (z = y → (x(φx = z) ↔ x(φx = y)))
135, 9, 12cbvex 1985 . 2 (zx(φx = z) ↔ yx(φx = y))
141, 13bitri 240 1 (∃!xφyx(φx = y))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  ∃!weu 2204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208 This theorem is referenced by:  eu1  2225  eumo0  2228
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