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Theorem 2reu4 41511
Description: Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2585. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
Assertion
Ref Expression
2reu4 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Distinct variable groups:   𝑧,𝑤,𝜑   𝑥,𝑤,𝑦,𝐴,𝑧   𝑤,𝐵,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2reu4
StepHypRef Expression
1 reurex 3190 . . . 4 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)
2 rexn0 4107 . . . 4 (∃𝑥𝐴𝑦𝐵 𝜑𝐴 ≠ ∅)
31, 2syl 17 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑𝐴 ≠ ∅)
4 reurex 3190 . . . 4 (∃!𝑦𝐵𝑥𝐴 𝜑 → ∃𝑦𝐵𝑥𝐴 𝜑)
5 rexn0 4107 . . . 4 (∃𝑦𝐵𝑥𝐴 𝜑𝐵 ≠ ∅)
64, 5syl 17 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑𝐵 ≠ ∅)
73, 6anim12i 589 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅))
8 ne0i 3954 . . . . . 6 (𝑥𝐴𝐴 ≠ ∅)
9 ne0i 3954 . . . . . 6 (𝑦𝐵𝐵 ≠ ∅)
108, 9anim12i 589 . . . . 5 ((𝑥𝐴𝑦𝐵) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅))
1110a1d 25 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝜑 → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)))
1211rexlimivv 3065 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅))
1312adantr 480 . 2 ((∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) → (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅))
14 2reu4a 41510 . 2 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))
157, 13, 14pm5.21nii 367 1 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2030  wne 2823  wral 2941  wrex 2942  ∃!wreu 2943  c0 3948
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-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-v 3233  df-dif 3610  df-nul 3949
This theorem is referenced by: (None)
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