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Theorem moim 2250
 Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
Assertion
Ref Expression
moim (x(φψ) → (∃*xψ∃*xφ))

Proof of Theorem moim
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 imim1 70 . . . 4 ((φψ) → ((ψx = y) → (φx = y)))
21al2imi 1561 . . 3 (x(φψ) → (x(ψx = y) → x(φx = y)))
32eximdv 1622 . 2 (x(φψ) → (yx(ψx = y) → yx(φx = y)))
4 nfv 1619 . . 3 yψ
54mo2 2233 . 2 (∃*xψyx(ψx = y))
6 nfv 1619 . . 3 yφ
76mo2 2233 . 2 (∃*xφyx(φx = y))
83, 5, 73imtr4g 261 1 (x(φψ) → (∃*xψ∃*xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃*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:  moimi  2251  euimmo  2253  moexex  2273  rmoim  3035  rmoimi2  3037  funmo  5125
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