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Theorem euf 1902
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 (φyφ)
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 1900 . 2 (∃!xφzx(φx = z))
2 euf.1 . . . . 5 (φyφ)
3 ax-17 1416 . . . . 5 (x = zy x = z)
42, 3hbbi 1437 . . . 4 ((φx = z) → y(φx = z))
54hbal 1363 . . 3 (x(φx = z) → yx(φx = z))
6 ax-17 1416 . . 3 (x(φx = y) → zx(φx = y))
7 equequ2 1596 . . . . 5 (z = y → (x = zx = y))
87bibi2d 221 . . . 4 (z = y → ((φx = z) ↔ (φx = y)))
98albidv 1702 . . 3 (z = y → (x(φx = z) ↔ x(φx = y)))
105, 6, 9cbvexh 1635 . 2 (zx(φx = z) ↔ yx(φx = y))
111, 10bitri 173 1 (∃!xφyx(φx = y))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240  wex 1378  ∃!weu 1897
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-eu 1900
This theorem is referenced by:  eu1  1922  eumo0  1928
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