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Theorem 2reu7 39840
Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2547. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Assertion
Ref Expression
2reu7 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem 2reu7
StepHypRef Expression
1 nfcv 2751 . . . 4 𝑥𝐵
2 nfre1 2988 . . . 4 𝑥𝑥𝐴 𝜑
31, 2nfreu 3093 . . 3 𝑥∃!𝑦𝐵𝑥𝐴 𝜑
43reuan 39829 . 2 (∃!𝑥𝐴 (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
5 ancom 465 . . . . 5 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑))
65reubii 3105 . . . 4 (∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑))
7 nfre1 2988 . . . . 5 𝑦𝑦𝐵 𝜑
87reuan 39829 . . . 4 (∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
9 ancom 465 . . . 4 ((∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
106, 8, 93bitri 285 . . 3 (∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
1110reubii 3105 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
12 ancom 465 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
134, 11, 123bitr4ri 292 1 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 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-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:  2reu8  39841
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