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Theorem reuun2 4076
Description: Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
Assertion
Ref Expression
reuun2 (¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reuun2
StepHypRef Expression
1 df-rex 3061 . . 3 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
2 euor2 2635 . . 3 (¬ ∃𝑥(𝑥𝐵𝜑) → (∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)) ↔ ∃!𝑥(𝑥𝐴𝜑)))
31, 2sylnbi 321 . 2 (¬ ∃𝑥𝐵 𝜑 → (∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)) ↔ ∃!𝑥(𝑥𝐴𝜑)))
4 df-reu 3062 . . 3 (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
5 elun 3917 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 617 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
7 andir 1031 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
8 orcom 896 . . . . . 6 (((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
97, 8bitri 266 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
106, 9bitri 266 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
1110eubii 2584 . . 3 (∃!𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
124, 11bitri 266 . 2 (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
13 df-reu 3062 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
143, 12, 133bitr4g 305 1 (¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  wex 1874  wcel 2155  ∃!weu 2581  wrex 3056  ∃!wreu 3057  cun 3732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-reu 3062  df-v 3352  df-un 3739
This theorem is referenced by:  hdmap14lem4a  37848
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