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Theorem moanim 2260
 Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)
Hypothesis
Ref Expression
moanim.1 xφ
Assertion
Ref Expression
moanim (∃*x(φ ψ) ↔ (φ∃*xψ))

Proof of Theorem moanim
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 impexp 433 . . . . 5 (((φ ψ) → x = y) ↔ (φ → (ψx = y)))
21albii 1566 . . . 4 (x((φ ψ) → x = y) ↔ x(φ → (ψx = y)))
3 moanim.1 . . . . 5 xφ
4319.21 1796 . . . 4 (x(φ → (ψx = y)) ↔ (φx(ψx = y)))
52, 4bitri 240 . . 3 (x((φ ψ) → x = y) ↔ (φx(ψx = y)))
65exbii 1582 . 2 (yx((φ ψ) → x = y) ↔ y(φx(ψx = y)))
7 nfv 1619 . . 3 y(φ ψ)
87mo2 2233 . 2 (∃*x(φ ψ) ↔ yx((φ ψ) → x = y))
9 nfv 1619 . . . . 5 yψ
109mo2 2233 . . . 4 (∃*xψyx(ψx = y))
1110imbi2i 303 . . 3 ((φ∃*xψ) ↔ (φyx(ψx = y)))
12 19.37v 1899 . . 3 (y(φx(ψx = y)) ↔ (φyx(ψx = y)))
1311, 12bitr4i 243 . 2 ((φ∃*xψ) ↔ y(φx(ψx = y)))
146, 8, 133bitr4i 268 1 (∃*x(φ ψ) ↔ (φ∃*xψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  ∃*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:  moanimv  2262  moaneu  2263  moanmo  2264  2eu1  2284
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