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Theorem eubidh 1903
Description: Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
Hypotheses
Ref Expression
eubidh.1 (φxφ)
eubidh.2 (φ → (ψχ))
Assertion
Ref Expression
eubidh (φ → (∃!xψ∃!xχ))

Proof of Theorem eubidh
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eubidh.1 . . . 4 (φxφ)
2 eubidh.2 . . . . 5 (φ → (ψχ))
32bibi1d 222 . . . 4 (φ → ((ψx = y) ↔ (χx = y)))
41, 3albidh 1366 . . 3 (φ → (x(ψx = y) ↔ x(χx = y)))
54exbidv 1703 . 2 (φ → (yx(ψx = y) ↔ yx(χx = y)))
6 df-eu 1900 . 2 (∃!xψyx(ψx = y))
7 df-eu 1900 . 2 (∃!xχyx(χx = y))
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
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
This theorem is referenced by:  euor  1923  mobidh  1931  euan  1953  euor2  1955  eupickbi  1979
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