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Theorem sb8mo 1911
Description: Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
sb8eu.1 yφ
Assertion
Ref Expression
sb8mo (∃*xφ∃*y[y / x]φ)

Proof of Theorem sb8mo
StepHypRef Expression
1 sb8eu.1 . . . 4 yφ
21sb8e 1734 . . 3 (xφy[y / x]φ)
31sb8eu 1910 . . 3 (∃!xφ∃!y[y / x]φ)
42, 3imbi12i 228 . 2 ((xφ∃!xφ) ↔ (y[y / x]φ∃!y[y / x]φ))
5 df-mo 1901 . 2 (∃*xφ ↔ (xφ∃!xφ))
6 df-mo 1901 . 2 (∃*y[y / x]φ ↔ (y[y / x]φ∃!y[y / x]φ))
74, 5, 63bitr4i 201 1 (∃*xφ∃*y[y / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wnf 1346  wex 1378  [wsb 1642  ∃!weu 1897  ∃*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-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by: (None)
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