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Theorem 2reu7 43662
Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2720. (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 2955 . . . 4 𝑥𝐵
2 nfre1 3265 . . . 4 𝑥𝑥𝐴 𝜑
31, 2nfreuw 3327 . . 3 𝑥∃!𝑦𝐵𝑥𝐴 𝜑
43reuan 3825 . 2 (∃!𝑥𝐴 (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
5 ancom 464 . . . . 5 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑))
65reubii 3344 . . . 4 (∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑))
7 nfre1 3265 . . . . 5 𝑦𝑦𝐵 𝜑
87reuan 3825 . . . 4 (∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
9 ancom 464 . . . 4 ((∃𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
106, 8, 93bitri 300 . . 3 (∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
1110reubii 3344 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
12 ancom 464 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
134, 11, 123bitr4ri 307 1 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wrex 3107  ∃!wreu 3108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2598  df-eu 2629  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114
This theorem is referenced by:  2reu8  43663
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