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Theorem 2reu8 40526
Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2559. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥𝐴∃!𝑦𝐵 using 2reu7 40525. (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 40521 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 → (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐵𝑥𝐴 𝜑))
21pm5.32i 668 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
3 nfcv 2761 . . . . 5 𝑥𝐵
4 nfreu1 3104 . . . . 5 𝑥∃!𝑥𝐴 𝜑
53, 4nfreu 3108 . . . 4 𝑥∃!𝑦𝐵 ∃!𝑥𝐴 𝜑
65reuan 40514 . . 3 (∃!𝑥𝐴 (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
7 ancom 466 . . . . . 6 ((∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃!𝑥𝐴 𝜑))
87reubii 3121 . . . . 5 (∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃!𝑥𝐴 𝜑))
9 nfre1 3001 . . . . . 6 𝑦𝑦𝐵 𝜑
109reuan 40514 . . . . 5 (∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃!𝑥𝐴 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑))
11 ancom 466 . . . . 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 3121 . . 3 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 (∃!𝑦𝐵 ∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
14 ancom 466 . . 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 40525 . 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 384  wrex 2909  ∃!wreu 2910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916
This theorem is referenced by: (None)
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