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Theorem reuun2 4051
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 3054 . . 3 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
2 euor2 2650 . . 3 (¬ ∃𝑥(𝑥𝐵𝜑) → (∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)) ↔ ∃!𝑥(𝑥𝐴𝜑)))
31, 2sylnbi 319 . 2 (¬ ∃𝑥𝐵 𝜑 → (∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)) ↔ ∃!𝑥(𝑥𝐴𝜑)))
4 df-reu 3055 . . 3 (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
5 elun 3894 . . . . . 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 2627 . . 3 (∃!𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
124, 11bitri 264 . 2 (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥((𝑥𝐵𝜑) ∨ (𝑥𝐴𝜑)))
13 df-reu 3055 . 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 1851  wcel 2137  ∃!weu 2605  wrex 3049  ∃!wreu 3050  cun 3711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-rex 3054  df-reu 3055  df-v 3340  df-un 3718
This theorem is referenced by:  hdmap14lem4a  37663
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