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Theorem eubidh 2009
 Description: Formula-building rule for unique existential quantifier (deduction form). (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 232 . . . 4 (𝜑 → ((𝜓𝑥 = 𝑦) ↔ (𝜒𝑥 = 𝑦)))
41, 3albidh 1457 . . 3 (𝜑 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜒𝑥 = 𝑦)))
54exbidv 1802 . 2 (𝜑 → (∃𝑦𝑥(𝜓𝑥 = 𝑦) ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦)))
6 df-eu 2006 . 2 (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦))
7 df-eu 2006 . 2 (∃!𝑥𝜒 ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦))
85, 6, 73bitr4g 222 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330  ∃wex 1469  ∃!weu 2003 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-17 1503  ax-ial 1511 This theorem depends on definitions:  df-bi 116  df-eu 2006 This theorem is referenced by:  euor  2029  mobidh  2037  euan  2059  euor2  2061  eupickbi  2085
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