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Theorem morimv 2252
 Description: Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
Assertion
Ref Expression
morimv (∃*x(φψ) → (φ∃*xψ))
Distinct variable group:   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem morimv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ax-1 6 . . . . . . 7 (ψ → (φψ))
21a1i 10 . . . . . 6 (φ → (ψ → (φψ)))
32imim1d 69 . . . . 5 (φ → (((φψ) → x = y) → (ψx = y)))
43alimdv 1621 . . . 4 (φ → (x((φψ) → x = y) → x(ψx = y)))
54eximdv 1622 . . 3 (φ → (yx((φψ) → x = y) → yx(ψx = y)))
6 nfv 1619 . . . 4 y(φψ)
76mo2 2233 . . 3 (∃*x(φψ) ↔ yx((φψ) → x = y))
8 nfv 1619 . . . 4 yψ
98mo2 2233 . . 3 (∃*xψyx(ψx = y))
105, 7, 93imtr4g 261 . 2 (φ → (∃*x(φψ) → ∃*xψ))
1110com12 27 1 (∃*x(φψ) → (φ∃*xψ))
 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: (None)
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