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Theorem mobid 1913
Description: Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
Hypotheses
Ref Expression
mobid.1 xφ
mobid.2 (φ → (ψχ))
Assertion
Ref Expression
mobid (φ → (∃*xψ∃*xχ))

Proof of Theorem mobid
StepHypRef Expression
1 mobid.1 . . . 4 xφ
2 mobid.2 . . . 4 (φ → (ψχ))
31, 2exbid 1485 . . 3 (φ → (xψxχ))
41, 2eubid 1885 . . 3 (φ → (∃!xψ∃!xχ))
53, 4imbi12d 223 . 2 (φ → ((xψ∃!xψ) ↔ (xχ∃!xχ)))
6 df-mo 1882 . 2 (∃*xψ ↔ (xψ∃!xψ))
7 df-mo 1882 . 2 (∃*xχ ↔ (xχ∃!xχ))
85, 6, 73bitr4g 212 1 (φ → (∃*xψ∃*xχ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wnf 1325  wex 1358  ∃!weu 1878  ∃*wmo 1879
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-5 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-17 1396  ax-ial 1405
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-eu 1881  df-mo 1882
This theorem is referenced by:  mobidv  1914  rmobida  2470  rmoeq1f  2478
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