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Theorem eubidh 2032
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 233 . . . 4 (𝜑 → ((𝜓𝑥 = 𝑦) ↔ (𝜒𝑥 = 𝑦)))
41, 3albidh 1480 . . 3 (𝜑 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜒𝑥 = 𝑦)))
54exbidv 1825 . 2 (𝜑 → (∃𝑦𝑥(𝜓𝑥 = 𝑦) ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦)))
6 df-eu 2029 . 2 (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦))
7 df-eu 2029 . 2 (∃!𝑥𝜒 ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦))
85, 6, 73bitr4g 223 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351  wex 1492  ∃!weu 2026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-eu 2029
This theorem is referenced by:  euor  2052  mobidh  2060  euan  2082  euor2  2084  eupickbi  2108
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