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Theorem euanv 2672
Description: Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
Assertion
Ref Expression
euanv (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem euanv
StepHypRef Expression
1 nfv 1992 . 2 𝑥𝜑
21euan 2668 1 (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  ∃!weu 2607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-eu 2611
This theorem is referenced by:  eueq2  3521  2reu5lem1  3554  fsn  6565  dfac5lem5  9140
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