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Theorem eu3 2230
 Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eu3.1 yφ
Assertion
Ref Expression
eu3 (∃!xφ ↔ (xφ yx(φx = y)))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem eu3
StepHypRef Expression
1 eu3.1 . . 3 yφ
21eu2 2229 . 2 (∃!xφ ↔ (xφ xy((φ [y / x]φ) → x = y)))
31mo 2226 . . 3 (yx(φx = y) ↔ xy((φ [y / x]φ) → x = y))
43anbi2i 675 . 2 ((xφ yx(φx = y)) ↔ (xφ xy((φ [y / x]φ) → x = y)))
52, 4bitr4i 243 1 (∃!xφ ↔ (xφ yx(φx = y)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  [wsb 1648  ∃!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 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208 This theorem is referenced by:  mo2  2233  eu5  2242  2eu4  2287  eqeu  3007  reu3  3026
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