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Theorem mo2 2233
 Description: Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.)
Hypothesis
Ref Expression
mo2.1 yφ
Assertion
Ref Expression
mo2 (∃*xφyx(φx = y))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem mo2
StepHypRef Expression
1 df-mo 2209 . 2 (∃*xφ ↔ (xφ∃!xφ))
2 alnex 1543 . . . . 5 (x ¬ φ ↔ ¬ xφ)
3 pm2.21 100 . . . . . . 7 φ → (φx = y))
43alimi 1559 . . . . . 6 (x ¬ φx(φx = y))
5 19.8a 1756 . . . . . 6 (x(φx = y) → yx(φx = y))
64, 5syl 15 . . . . 5 (x ¬ φyx(φx = y))
72, 6sylbir 204 . . . 4 xφyx(φx = y))
8 mo2.1 . . . . 5 yφ
98eumo0 2228 . . . 4 (∃!xφyx(φx = y))
107, 9ja 153 . . 3 ((xφ∃!xφ) → yx(φx = y))
118eu3 2230 . . . 4 (∃!xφ ↔ (xφ yx(φx = y)))
1211simplbi2com 1374 . . 3 (yx(φx = y) → (xφ∃!xφ))
1310, 12impbii 180 . 2 ((xφ∃!xφ) ↔ yx(φx = y))
141, 13bitri 240 1 (∃*xφyx(φx = y))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  ∃!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:  sbmo  2234  mo3  2235  eu5  2242  moim  2250  morimv  2252  moanim  2260  mo2icl  3015  rmo2  3131  dffun3  5120  dffun6f  5123
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