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Theorem 2rexreu 41950
Description: Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2703. (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 3342 . . . 4 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴𝑦𝐵 𝜑)
2 reurex 3341 . . . . 5 (∃!𝑦𝐵 𝜑 → ∃𝑦𝐵 𝜑)
32rmoimi 41941 . . . 4 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
41, 3syl 17 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
5 2reurex 41946 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 → ∃𝑥𝐴 ∃!𝑦𝐵 𝜑)
64, 5anim12ci 608 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
7 reu5 3340 . 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 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  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095
This theorem is referenced by:  2reu1  41951  2reu2  41952  2reu3  41953
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