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Theorem 2reu7 41697
Description: Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2697. (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 2902 . . . 4 𝑥𝐵
2 nfre1 3143 . . . 4 𝑥𝑥𝐴 𝜑
31, 2nfreu 3252 . . 3 𝑥∃!𝑦𝐵𝑥𝐴 𝜑
43reuan 41686 . 2 (∃!𝑥𝐴 (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃!𝑥𝐴𝑦𝐵 𝜑))
5 ancom 465 . . . . 5 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑))
65reubii 3267 . . . 4 (∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑦𝐵 (∃𝑦𝐵 𝜑 ∧ ∃𝑥𝐴 𝜑))
7 nfre1 3143 . . . . 5 𝑦𝑦𝐵 𝜑
87reuan 41686 . . . 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 286 . . 3 (∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ (∃!𝑦𝐵𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
1110reubii 3267 . 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 293 1 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wrex 3051  ∃!wreu 3052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-eu 2611  df-mo 2612  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058
This theorem is referenced by:  2reu8  41698
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