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Theorem 2reu2 3854
Description: Double restricted existential uniqueness, analogous to 2eu2 2738. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
Assertion
Ref Expression
2reu2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem 2reu2
StepHypRef Expression
1 reurmo 3406 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑)
2 2rmorex 3720 . . 3 (∃*𝑦𝐵𝑥𝐴 𝜑 → ∀𝑥𝐴 ∃*𝑦𝐵 𝜑)
3 2reu1 3853 . . . 4 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))
4 simpl 486 . . . 4 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴𝑦𝐵 𝜑)
53, 4syl6bi 256 . . 3 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑))
61, 2, 53syl 18 . 2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑))
7 2rexreu 3728 . . 3 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
87expcom 417 . 2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑))
96, 8impbid 215 1 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wral 3130  wrex 3131  ∃!wreu 3132  ∃*wrmo 3133
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138
This theorem is referenced by:  2reu8  43607
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