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Theorem reuun2 4213
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))

Proof of Theorem reuun2
StepHypRef Expression
1 df-rex 3060 . . 3 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
2 euor2 2617 . . 3 (¬ ∃𝑥(𝑥𝐵𝜑) → (∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)) ↔ ∃!𝑥(𝑥𝐴𝜑)))
31, 2sylnbi 333 . 2 (¬ ∃𝑥𝐵 𝜑 → (∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)) ↔ ∃!𝑥(𝑥𝐴𝜑)))
4 df-reu 3061 . . 3 (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
5 elun 4049 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 627 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
7 andir 1008 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
8 orcom 869 . . . . . 6 (((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
97, 8bitri 278 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
106, 9bitri 278 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
1110eubii 2587 . . 3 (∃!𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
124, 11bitri 278 . 2 (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
13 df-reu 3061 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
143, 12, 133bitr4g 317 1 (¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846  wex 1786  wcel 2114  ∃!weu 2570  wrex 3055  ∃!wreu 3056  cun 3851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-rex 3060  df-reu 3061  df-v 3402  df-un 3858
This theorem is referenced by:  hdmap14lem4a  39541
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