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Theorem eubid 1904
 Description: Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
Hypotheses
Ref Expression
eubid.1 xφ
eubid.2 (φ → (ψχ))
Assertion
Ref Expression
eubid (φ → (∃!xψ∃!xχ))

Proof of Theorem eubid
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eubid.1 . . . 4 xφ
2 eubid.2 . . . . 5 (φ → (ψχ))
32bibi1d 222 . . . 4 (φ → ((ψx = y) ↔ (χx = y)))
41, 3albid 1503 . . 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  Ⅎwnf 1346  ∃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-nf 1347  df-eu 1900 This theorem is referenced by:  eubidv  1905  mobid  1932  reubida  2485  reueq1f  2497  eusv2i  4153
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