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Theorem eu4 2243
 Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
Hypothesis
Ref Expression
eu4.1 (x = y → (φψ))
Assertion
Ref Expression
eu4 (∃!xφ ↔ (xφ xy((φ ψ) → x = y)))
Distinct variable groups:   x,y   φ,y   ψ,x
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem eu4
StepHypRef Expression
1 eu5 2242 . 2 (∃!xφ ↔ (xφ ∃*xφ))
2 eu4.1 . . . 4 (x = y → (φψ))
32mo4 2237 . . 3 (∃*xφxy((φ ψ) → x = y))
43anbi2i 675 . 2 ((xφ ∃*xφ) ↔ (xφ xy((φ ψ) → x = y)))
51, 4bitri 240 1 (∃!xφ ↔ (xφ xy((φ ψ) → x = y)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205 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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209 This theorem is referenced by:  euequ1  2292  eueq  3008  euind  3023  uniintsn  3963
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