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Theorem 2reu5 3549
Description: Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2687 and reu3 3529. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2reu5 ((∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∀𝑥𝐴 ∃*𝑦𝐵 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Distinct variable groups:   𝑦,𝑤,𝑧,𝐴,𝑥   𝑤,𝐵   𝑥,𝑧,𝐵,𝑦   𝜑,𝑤,𝑧   𝑥,𝐴   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2reu5
StepHypRef Expression
1 r19.29r 3203 . . . . . . . 8 ((∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) → ∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
2 r19.29r 3203 . . . . . . . . 9 ((∃𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) → ∃𝑦𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
32reximi 3141 . . . . . . . 8 (∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) → ∃𝑥𝐴𝑦𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
4 pm3.35 612 . . . . . . . . . 10 ((𝜑 ∧ (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) → (𝑥 = 𝑧𝑦 = 𝑤))
54reximi 3141 . . . . . . . . 9 (∃𝑦𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) → ∃𝑦𝐵 (𝑥 = 𝑧𝑦 = 𝑤))
65reximi 3141 . . . . . . . 8 (∃𝑥𝐴𝑦𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) → ∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑧𝑦 = 𝑤))
7 eleq1w 2814 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
8 eleq1w 2814 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
97, 8bi2anan9 953 . . . . . . . . . . . 12 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝐵)))
109biimpac 504 . . . . . . . . . . 11 (((𝑥𝐴𝑦𝐵) ∧ (𝑥 = 𝑧𝑦 = 𝑤)) → (𝑧𝐴𝑤𝐵))
1110ancomd 466 . . . . . . . . . 10 (((𝑥𝐴𝑦𝐵) ∧ (𝑥 = 𝑧𝑦 = 𝑤)) → (𝑤𝐵𝑧𝐴))
1211ex 449 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑤𝐵𝑧𝐴)))
1312rexlimivv 3166 . . . . . . . 8 (∃𝑥𝐴𝑦𝐵 (𝑥 = 𝑧𝑦 = 𝑤) → (𝑤𝐵𝑧𝐴))
141, 3, 6, 134syl 19 . . . . . . 7 ((∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) → (𝑤𝐵𝑧𝐴))
1514ex 449 . . . . . 6 (∃𝑥𝐴𝑦𝐵 𝜑 → (∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝑤𝐵𝑧𝐴)))
1615pm4.71rd 670 . . . . 5 (∃𝑥𝐴𝑦𝐵 𝜑 → (∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝑤𝐵𝑧𝐴) ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))
17 anass 684 . . . . 5 (((𝑤𝐵𝑧𝐴) ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) ↔ (𝑤𝐵 ∧ (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))
1816, 17syl6bb 276 . . . 4 (∃𝑥𝐴𝑦𝐵 𝜑 → (∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (𝑤𝐵 ∧ (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))))
19182exbidv 1993 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 → (∃𝑧𝑤𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑤(𝑤𝐵 ∧ (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))))
2019pm5.32i 672 . 2 ((∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝑤𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝑤(𝑤𝐵 ∧ (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))))
21 2reu5lem3 3548 . 2 ((∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∀𝑥𝐴 ∃*𝑦𝐵 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝑤𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
22 df-rex 3048 . . . 4 (∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧(𝑧𝐴 ∧ ∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
23 r19.42v 3222 . . . . . 6 (∃𝑤𝐵 (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) ↔ (𝑧𝐴 ∧ ∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
24 df-rex 3048 . . . . . 6 (∃𝑤𝐵 (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) ↔ ∃𝑤(𝑤𝐵 ∧ (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))
2523, 24bitr3i 266 . . . . 5 ((𝑧𝐴 ∧ ∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) ↔ ∃𝑤(𝑤𝐵 ∧ (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))
2625exbii 1915 . . . 4 (∃𝑧(𝑧𝐴 ∧ ∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) ↔ ∃𝑧𝑤(𝑤𝐵 ∧ (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))
2722, 26bitri 264 . . 3 (∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∃𝑧𝑤(𝑤𝐵 ∧ (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))
2827anbi2i 732 . 2 ((∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝑤(𝑤𝐵 ∧ (𝑧𝐴 ∧ ∀𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))))
2920, 21, 283bitr4i 292 1 ((∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∀𝑥𝐴 ∃*𝑦𝐵 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wex 1845  wcel 2131  wral 3042  wrex 3043  ∃!wreu 3044  ∃*wrmo 3045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-eu 2603  df-mo 2604  df-clel 2748  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050
This theorem is referenced by: (None)
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