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Theorem 2reu8 39841
 Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2548. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵 using 2reu7 39840. (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 39836 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 → (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐵𝑥𝐴 𝜑))
21pm5.32i 667 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
3 nfcv 2751 . . . . 5 𝑥𝐵
4 nfreu1 3089 . . . . 5 𝑥∃!𝑥𝐴 𝜑
53, 4nfreu 3093 . . . 4 𝑥∃!𝑦𝐵 ∃!𝑥𝐴 𝜑
65reuan 39829 . . 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 3105 . . . . 5 (∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃!𝑥𝐴 𝜑))
9 nfre1 2988 . . . . . 6 𝑦𝑦𝐵 𝜑
109reuan 39829 . . . . 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 285 . . . 4 (∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
1312reubii 3105 . . 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 292 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
16 2reu7 39840 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
172, 15, 163bitr3ri 290 1 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wrex 2897  ∃!wreu 2898 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904 This theorem is referenced by: (None)
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