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Theorem eubid 2211
 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 310 . . . 4 (φ → ((ψx = y) ↔ (χx = y)))
41, 3albid 1772 . . 3 (φ → (x(ψx = y) ↔ x(χx = y)))
54exbidv 1626 . 2 (φ → (yx(ψx = y) ↔ yx(χx = y)))
6 df-eu 2208 . 2 (∃!xψyx(ψx = y))
7 df-eu 2208 . 2 (∃!xχyx(χx = y))
85, 6, 73bitr4g 279 1 (φ → (∃!xψ∃!xχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  ∃!weu 2204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-eu 2208 This theorem is referenced by:  eubidv  2212  euor  2231  mobid  2238  euan  2261  eupickbi  2270  euor2  2272  reubida  2793  reueq1f  2805
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