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Theorem euanv 2522
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 1830 . 2 𝑥𝜑
21euan 2518 1 (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  ∃!weu 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-eu 2462
This theorem is referenced by:  eueq2  3347  2reu5lem1  3380  fsn  6308  dfac5lem5  8833
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