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Theorem reuun2 4054
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 3057 . . 3 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
2 euor2 2653 . . 3 (¬ ∃𝑥(𝑥𝐵𝜑) → (∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)) ↔ ∃!𝑥(𝑥𝐴𝜑)))
31, 2sylnbi 319 . 2 (¬ ∃𝑥𝐵 𝜑 → (∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)) ↔ ∃!𝑥(𝑥𝐴𝜑)))
4 df-reu 3058 . . 3 (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
5 elun 3897 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 733 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
7 andir 948 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)))
8 orcom 401 . . . . . 6 (((𝑥𝐴𝜑) ∨ (𝑥𝐵𝜑)) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
97, 8bitri 264 . . . . 5 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
106, 9bitri 264 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
1110eubii 2630 . . 3 (∃!𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
124, 11bitri 264 . 2 (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
13 df-reu 3058 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
143, 12, 133bitr4g 303 1 (¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  wex 1853  wcel 2140  ∃!weu 2608  wrex 3052  ∃!wreu 3053  cun 3714
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 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rex 3057  df-reu 3058  df-v 3343  df-un 3721
This theorem is referenced by:  hdmap14lem4a  37684
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