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Theorem 2reu5a 41942
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 3340 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
2 reu5 3340 . . . 4 (∃!𝑦𝐵 𝜑 ↔ (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑))
32rexbii 3220 . . 3 (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑))
42rmobii 3314 . . 3 (∃*𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃*𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑))
53, 4anbi12i 621 . 2 ((∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑) ↔ (∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑) ∧ ∃*𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑)))
61, 5bitri 267 1 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑) ∧ ∃*𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  wrex 3088  ∃!wreu 3089  ∃*wrmo 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-mo 2590  df-eu 2607  df-rex 3093  df-reu 3094  df-rmo 3095
This theorem is referenced by:  2reu1  41951
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