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Theorem eubidh 1906
 Description: Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
Hypotheses
Ref Expression
eubidh.1 (𝜑 → ∀𝑥𝜑)
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 (𝜑 → ∀𝑥𝜑)
2 eubidh.2 . . . . 5 (𝜑 → (𝜓𝜒))
32bibi1d 222 . . . 4 (𝜑 → ((𝜓𝑥 = 𝑦) ↔ (𝜒𝑥 = 𝑦)))
41, 3albidh 1369 . . 3 (𝜑 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜒𝑥 = 𝑦)))
54exbidv 1706 . 2 (𝜑 → (∃𝑦𝑥(𝜓𝑥 = 𝑦) ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦)))
6 df-eu 1903 . 2 (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦))
7 df-eu 1903 . 2 (∃!𝑥𝜒 ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦))
85, 6, 73bitr4g 212 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381  ∃!weu 1900 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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-eu 1903 This theorem is referenced by:  euor  1926  mobidh  1934  euan  1956  euor2  1958  eupickbi  1982
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