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Theorem mobidh 1931
Description: Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
Hypotheses
Ref Expression
mobidh.1 (φxφ)
mobidh.2 (φ → (ψχ))
Assertion
Ref Expression
mobidh (φ → (∃*xψ∃*xχ))

Proof of Theorem mobidh
StepHypRef Expression
1 mobidh.1 . . . 4 (φxφ)
2 mobidh.2 . . . 4 (φ → (ψχ))
31, 2exbidh 1502 . . 3 (φ → (xψxχ))
41, 2eubidh 1903 . . 3 (φ → (∃!xψ∃!xχ))
53, 4imbi12d 223 . 2 (φ → ((xψ∃!xψ) ↔ (xχ∃!xχ)))
6 df-mo 1901 . 2 (∃*xψ ↔ (xψ∃!xψ))
7 df-mo 1901 . 2 (∃*xχ ↔ (xχ∃!xχ))
85, 6, 73bitr4g 212 1 (φ → (∃*xψ∃*xχ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240  wex 1378  ∃!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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-eu 1900  df-mo 1901
This theorem is referenced by:  euan  1953
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