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Theorem eubid 2673
Description: Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 19-Feb-2023.)
Hypotheses
Ref Expression
eubid.1 𝑥𝜑
eubid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
eubid (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Proof of Theorem eubid
StepHypRef Expression
1 eubid.1 . . 3 𝑥𝜑
2 eubid.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimi 2214 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 eubi 2669 . 2 (∀𝑥(𝜓𝜒) → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
53, 4syl 17 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wnf 1785  ∃!weu 2653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-mo 2623  df-eu 2654
This theorem is referenced by:  euor  2695  euor2  2697  euan  2706  reubida  3372  reueq1f  3384  eusv2i  5268  reusv2lem3  5274  eubiOLD  40923
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