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Theorem moimv 1963
Description: Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
Assertion
Ref Expression
moimv (∃*x(φψ) → (φ∃*xψ))
Distinct variable group:   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem moimv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ax-1 5 . . . . . . 7 (ψ → (φψ))
21a1i 9 . . . . . 6 (φ → (ψ → (φψ)))
32sbimi 1644 . . . . . . 7 ([y / x]φ → [y / x](ψ → (φψ)))
4 nfv 1418 . . . . . . . 8 xφ
54sbf 1657 . . . . . . 7 ([y / x]φφ)
6 sbim 1824 . . . . . . 7 ([y / x](ψ → (φψ)) ↔ ([y / x]ψ → [y / x](φψ)))
73, 5, 63imtr3i 189 . . . . . 6 (φ → ([y / x]ψ → [y / x](φψ)))
82, 7anim12d 318 . . . . 5 (φ → ((ψ [y / x]ψ) → ((φψ) [y / x](φψ))))
98imim1d 69 . . . 4 (φ → ((((φψ) [y / x](φψ)) → x = y) → ((ψ [y / x]ψ) → x = y)))
1092alimdv 1758 . . 3 (φ → (xy(((φψ) [y / x](φψ)) → x = y) → xy((ψ [y / x]ψ) → x = y)))
11 ax-17 1416 . . . 4 ((φψ) → y(φψ))
1211mo3h 1950 . . 3 (∃*x(φψ) ↔ xy(((φψ) [y / x](φψ)) → x = y))
13 ax-17 1416 . . . 4 (ψyψ)
1413mo3h 1950 . . 3 (∃*xψxy((ψ [y / x]ψ) → x = y))
1510, 12, 143imtr4g 194 . 2 (φ → (∃*x(φψ) → ∃*xψ))
1615com12 27 1 (∃*x(φψ) → (φ∃*xψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240  [wsb 1642  ∃*wmo 1898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by: (None)
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