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Theorem 2reu5a 40946
Description: Double restricted existential uniqueness in terms of restricted existence and restricted "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2reu5a (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑) ∧ ∃*𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑)))

Proof of Theorem 2reu5a
StepHypRef Expression
1 reu5 3157 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
2 reu5 3157 . . . 4 (∃!𝑦𝐵 𝜑 ↔ (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑))
32rexbii 3039 . . 3 (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑))
42rmobii 3131 . . 3 (∃*𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃*𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑))
53, 4anbi12i 733 . 2 ((∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑) ↔ (∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑) ∧ ∃*𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑)))
61, 5bitri 264 1 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑) ∧ ∃*𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wrex 2912  ∃!wreu 2913  ∃*wrmo 2914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-eu 2473  df-mo 2474  df-rex 2917  df-reu 2918  df-rmo 2919
This theorem is referenced by:  2reu1  40955
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