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Theorem reuanid 3067
Description: Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018.)
Assertion
Ref Expression
reuanid (∃!𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃!𝑥𝐴 𝜑)

Proof of Theorem reuanid
StepHypRef Expression
1 anabs5 653 . . 3 ((𝑥𝐴 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴𝜑))
21eubii 2584 . 2 (∃!𝑥(𝑥𝐴 ∧ (𝑥𝐴𝜑)) ↔ ∃!𝑥(𝑥𝐴𝜑))
3 df-reu 3062 . 2 (∃!𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐴 ∧ (𝑥𝐴𝜑)))
4 df-reu 3062 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
52, 3, 43bitr4i 294 1 (∃!𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wcel 2155  ∃!weu 2581  ∃!wreu 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-mo 2565  df-eu 2582  df-reu 3062
This theorem is referenced by: (None)
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