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Theorem euanv 2521
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 1829 . 2 𝑥𝜑
21euan 2517 1 (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  ∃!weu 2457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1700  df-eu 2461
This theorem is referenced by:  eueq2  3346  2reu5lem1  3379  fsn  6292  dfac5lem5  8810
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