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Theorem 2rexreu 3768
Description: Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2646. (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 3383 . . . 4 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴𝑦𝐵 𝜑)
2 reurex 3384 . . . . 5 (∃!𝑦𝐵 𝜑 → ∃𝑦𝐵 𝜑)
32rmoimi 3748 . . . 4 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
41, 3syl 17 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
5 2reurex 3766 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 → ∃𝑥𝐴 ∃!𝑦𝐵 𝜑)
64, 5anim12ci 614 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
7 reu5 3382 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
86, 7sylibr 234 1 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wrex 3070  ∃!wreu 3378  ∃*wrmo 3379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-10 2141  ax-11 2157  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2540  df-eu 2569  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381
This theorem is referenced by:  2reu1  3897  2reu2  3898  opreu2reu  6315  2reu2reu2  32502  2reu3  47122
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