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Theorem 2reu2 42012
 Description: Double restricted existential uniqueness, analogous to 2eu2 2734. (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 3373 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑)
2 2rmorex 3639 . . 3 (∃*𝑦𝐵𝑥𝐴 𝜑 → ∀𝑥𝐴 ∃*𝑦𝐵 𝜑)
3 2reu1 42011 . . . 4 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))
4 simpl 476 . . . 4 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴𝑦𝐵 𝜑)
53, 4syl6bi 245 . . 3 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑))
61, 2, 53syl 18 . 2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑))
7 2rexreu 42010 . . 3 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
87expcom 404 . 2 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑))
96, 8impbid 204 1 (∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wral 3117  ∃wrex 3118  ∃!wreu 3119  ∃*wrmo 3120 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125 This theorem is referenced by:  2reu8  42017
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