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Theorem reuanid 3333
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 659 . . 3 ((𝑥𝐴 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴𝜑))
21eubii 2668 . 2 (∃!𝑥(𝑥𝐴 ∧ (𝑥𝐴𝜑)) ↔ ∃!𝑥(𝑥𝐴𝜑))
3 df-reu 3150 . 2 (∃!𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐴 ∧ (𝑥𝐴𝜑)))
4 df-reu 3150 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
52, 3, 43bitr4i 304 1 (∃!𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2107  ∃!weu 2651  ∃!wreu 3145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-mo 2620  df-eu 2652  df-reu 3150
This theorem is referenced by: (None)
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