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Theorem 2rexreu 41705
Description: Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2698. (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 3310 . . . 4 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴𝑦𝐵 𝜑)
2 reurex 3309 . . . . 5 (∃!𝑦𝐵 𝜑 → ∃𝑦𝐵 𝜑)
32rmoimi 41696 . . . 4 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
41, 3syl 17 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
5 2reurex 41701 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 → ∃𝑥𝐴 ∃!𝑦𝐵 𝜑)
64, 5anim12ci 601 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
7 reu5 3308 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
86, 7sylibr 224 1 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wrex 3062  ∃!wreu 3063  ∃*wrmo 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069
This theorem is referenced by:  2reu1  41706  2reu2  41707  2reu3  41708
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