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Theorem 2reu8 41513
Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2589. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥𝐴∃!𝑦𝐵 using 2reu7 41512. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Assertion
Ref Expression
2reu8 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2reu8
StepHypRef Expression
1 2reu2 41508 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 → (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐵𝑥𝐴 𝜑))
21pm5.32i 670 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
3 nfcv 2793 . . . . 5 𝑥𝐵
4 nfreu1 3139 . . . . 5 𝑥∃!𝑥𝐴 𝜑
53, 4nfreu 3143 . . . 4 𝑥∃!𝑦𝐵 ∃!𝑥𝐴 𝜑
65reuan 41501 . . 3 (∃!𝑥𝐴 (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
7 ancom 465 . . . . . 6 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃!𝑥𝐴 𝜑))
87reubii 3158 . . . . 5 (∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃!𝑥𝐴 𝜑))
9 nfre1 3034 . . . . . 6 𝑦𝑦𝐵 𝜑
109reuan 41501 . . . . 5 (∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃!𝑥𝐴 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑))
11 ancom 465 . . . . 5 ((∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
128, 10, 113bitri 286 . . . 4 (∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
1312reubii 3158 . . 3 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
14 ancom 465 . . 3 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
156, 13, 143bitr4ri 293 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
16 2reu7 41512 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
172, 15, 163bitr3ri 291 1 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wrex 2942  ∃!wreu 2943
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: (None)
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