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Theorem reuun2 3538
 Description: Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
Assertion
Ref Expression
reuun2 x B φ → (∃!x (AB)φ∃!x A φ))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem reuun2
StepHypRef Expression
1 df-rex 2620 . . 3 (x B φx(x B φ))
2 euor2 2272 . . 3 x(x B φ) → (∃!x((x B φ) (x A φ)) ↔ ∃!x(x A φ)))
31, 2sylnbi 297 . 2 x B φ → (∃!x((x B φ) (x A φ)) ↔ ∃!x(x A φ)))
4 df-reu 2621 . . 3 (∃!x (AB)φ∃!x(x (AB) φ))
5 elun 3220 . . . . . 6 (x (AB) ↔ (x A x B))
65anbi1i 676 . . . . 5 ((x (AB) φ) ↔ ((x A x B) φ))
7 andir 838 . . . . . 6 (((x A x B) φ) ↔ ((x A φ) (x B φ)))
8 orcom 376 . . . . . 6 (((x A φ) (x B φ)) ↔ ((x B φ) (x A φ)))
97, 8bitri 240 . . . . 5 (((x A x B) φ) ↔ ((x B φ) (x A φ)))
106, 9bitri 240 . . . 4 ((x (AB) φ) ↔ ((x B φ) (x A φ)))
1110eubii 2213 . . 3 (∃!x(x (AB) φ) ↔ ∃!x((x B φ) (x A φ)))
124, 11bitri 240 . 2 (∃!x (AB)φ∃!x((x B φ) (x A φ)))
13 df-reu 2621 . 2 (∃!x A φ∃!x(x A φ))
143, 12, 133bitr4g 279 1 x B φ → (∃!x (AB)φ∃!x A φ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃wrex 2615  ∃!wreu 2616   ∪ cun 3207 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-reu 2621  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214 This theorem is referenced by: (None)
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