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Theorem 2reu2 41508
Description: Double restricted existential uniqueness, analogous to 2eu2 2583. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
Assertion
Ref Expression
2reu2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem 2reu2
StepHypRef Expression
1 reurmo 3191 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑)
2 2rmorex 3445 . . 3 (∃*𝑦𝐵𝑥𝐴 𝜑 → ∀𝑥𝐴 ∃*𝑦𝐵 𝜑)
3 2reu1 41507 . . . 4 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))
4 simpl 472 . . . 4 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴𝑦𝐵 𝜑)
53, 4syl6bi 243 . . 3 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑))
61, 2, 53syl 18 . 2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑))
7 2rexreu 41506 . . 3 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
87expcom 450 . 2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑))
96, 8impbid 202 1 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wral 2941  wrex 2942  ∃!wreu 2943  ∃*wrmo 2944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949
This theorem is referenced by:  2reu8  41513
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