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Theorem mo4f 2236
 Description: "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
Hypotheses
Ref Expression
mo4f.1 xψ
mo4f.2 (x = y → (φψ))
Assertion
Ref Expression
mo4f (∃*xφxy((φ ψ) → x = y))
Distinct variable groups:   x,y   φ,y
Allowed substitution hints:   φ(x)   ψ(x,y)

Proof of Theorem mo4f
StepHypRef Expression
1 nfv 1619 . . 3 yφ
21mo3 2235 . 2 (∃*xφxy((φ [y / x]φ) → x = y))
3 mo4f.1 . . . . . 6 xψ
4 mo4f.2 . . . . . 6 (x = y → (φψ))
53, 4sbie 2038 . . . . 5 ([y / x]φψ)
65anbi2i 675 . . . 4 ((φ [y / x]φ) ↔ (φ ψ))
76imbi1i 315 . . 3 (((φ [y / x]φ) → x = y) ↔ ((φ ψ) → x = y))
872albii 1567 . 2 (xy((φ [y / x]φ) → x = y) ↔ xy((φ ψ) → x = y))
92, 8bitri 240 1 (∃*xφxy((φ ψ) → x = y))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544  [wsb 1648  ∃*wmo 2205 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  df-mo 2209 This theorem is referenced by:  mo4  2237  mob2  3016
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