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Theorem 2rexreu 42011
 Description: Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2730. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
2rexreu ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2rexreu
StepHypRef Expression
1 reurmo 3374 . . . 4 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴𝑦𝐵 𝜑)
2 reurex 3373 . . . . 5 (∃!𝑦𝐵 𝜑 → ∃𝑦𝐵 𝜑)
32rmoimi 42002 . . . 4 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
41, 3syl 17 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
5 2reurex 42007 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 → ∃𝑥𝐴 ∃!𝑦𝐵 𝜑)
64, 5anim12ci 609 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
7 reu5 3372 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
86, 7sylibr 226 1 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386  ∃wrex 3119  ∃!wreu 3120  ∃*wrmo 3121 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126 This theorem is referenced by:  2reu1  42012  2reu2  42013  2reu3  42014
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