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Theorem eubid 2476
 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 332 . . . 4 (𝜑 → ((𝜓𝑥 = 𝑦) ↔ (𝜒𝑥 = 𝑦)))
41, 3albid 2077 . . 3 (𝜑 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜒𝑥 = 𝑦)))
54exbidv 1837 . 2 (𝜑 → (∃𝑦𝑥(𝜓𝑥 = 𝑦) ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦)))
6 df-eu 2462 . 2 (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦))
7 df-eu 2462 . 2 (∃!𝑥𝜒 ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦))
85, 6, 73bitr4g 302 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699  ∃!weu 2458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701  df-eu 2462 This theorem is referenced by:  mobid  2477  eubidv  2478  euor  2500  euor2  2502  euan  2518  reubida  3101  reueq1f  3113  eusv2i  4789  reusv2lem3  4797  eubi  37659  nbusgredgeu0  40596
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