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Theorem 2reu7 40954
Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2557. (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 2762 . . . 4 𝑥𝐵
2 nfre1 3002 . . . 4 𝑥𝑥𝐴 𝜑
31, 2nfreu 3109 . . 3 𝑥∃!𝑦𝐵𝑥𝐴 𝜑
43reuan 40943 . 2 (∃!𝑥𝐴 (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
5 ancom 466 . . . . 5 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑))
65reubii 3123 . . . 4 (∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑))
7 nfre1 3002 . . . . 5 𝑦𝑦𝐵 𝜑
87reuan 40943 . . . 4 (∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
9 ancom 466 . . . 4 ((∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
106, 8, 93bitri 286 . . 3 (∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
1110reubii 3123 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
12 ancom 466 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
134, 11, 123bitr4ri 293 1 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wrex 2910  ∃!wreu 2911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-eu 2472  df-mo 2473  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917
This theorem is referenced by:  2reu8  40955
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