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Theorem reu3 2725
Description: A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)
Assertion
Ref Expression
reu3 (∃!x A φ ↔ (x A φ y A x A (φx = y)))
Distinct variable groups:   x,y,A   φ,y
Allowed substitution hint:   φ(x)

Proof of Theorem reu3
StepHypRef Expression
1 reurex 2517 . . 3 (∃!x A φx A φ)
2 reu6 2724 . . . 4 (∃!x A φy A x A (φx = y))
3 bi1 111 . . . . . 6 ((φx = y) → (φx = y))
43ralimi 2378 . . . . 5 (x A (φx = y) → x A (φx = y))
54reximi 2410 . . . 4 (y A x A (φx = y) → y A x A (φx = y))
62, 5sylbi 114 . . 3 (∃!x A φy A x A (φx = y))
71, 6jca 290 . 2 (∃!x A φ → (x A φ y A x A (φx = y)))
8 rexex 2362 . . . 4 (y A x A (φx = y) → yx A (φx = y))
98anim2i 324 . . 3 ((x A φ y A x A (φx = y)) → (x A φ yx A (φx = y)))
10 nfv 1418 . . . . 5 y(x A φ)
1110eu3 1943 . . . 4 (∃!x(x A φ) ↔ (x(x A φ) yx((x A φ) → x = y)))
12 df-reu 2307 . . . 4 (∃!x A φ∃!x(x A φ))
13 df-rex 2306 . . . . 5 (x A φx(x A φ))
14 df-ral 2305 . . . . . . 7 (x A (φx = y) ↔ x(x A → (φx = y)))
15 impexp 250 . . . . . . . 8 (((x A φ) → x = y) ↔ (x A → (φx = y)))
1615albii 1356 . . . . . . 7 (x((x A φ) → x = y) ↔ x(x A → (φx = y)))
1714, 16bitr4i 176 . . . . . 6 (x A (φx = y) ↔ x((x A φ) → x = y))
1817exbii 1493 . . . . 5 (yx A (φx = y) ↔ yx((x A φ) → x = y))
1913, 18anbi12i 433 . . . 4 ((x A φ yx A (φx = y)) ↔ (x(x A φ) yx((x A φ) → x = y)))
2011, 12, 193bitr4i 201 . . 3 (∃!x A φ ↔ (x A φ yx A (φx = y)))
219, 20sylibr 137 . 2 ((x A φ y A x A (φx = y)) → ∃!x A φ)
227, 21impbii 117 1 (∃!x A φ ↔ (x A φ y A x A (φx = y)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378   wcel 1390  ∃!weu 1897  wral 2300  wrex 2301  ∃!wreu 2302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308
This theorem is referenced by:  reu7  2730  bdreu  9310
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