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Theorem reu2 2723
Description: A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
reu2 (∃!x A φ ↔ (x A φ x A y A ((φ [y / x]φ) → x = y)))
Distinct variable groups:   x,y,A   φ,y
Allowed substitution hint:   φ(x)

Proof of Theorem reu2
StepHypRef Expression
1 nfv 1418 . . 3 y(x A φ)
21eu2 1941 . 2 (∃!x(x A φ) ↔ (x(x A φ) xy(((x A φ) [y / x](x A φ)) → x = y)))
3 df-reu 2307 . 2 (∃!x A φ∃!x(x A φ))
4 df-rex 2306 . . 3 (x A φx(x A φ))
5 df-ral 2305 . . . 4 (x A y A ((φ [y / x]φ) → x = y) ↔ x(x Ay A ((φ [y / x]φ) → x = y)))
6 19.21v 1750 . . . . . 6 (y(x A → (y A → ((φ [y / x]φ) → x = y))) ↔ (x Ay(y A → ((φ [y / x]φ) → x = y))))
7 nfv 1418 . . . . . . . . . . . . 13 x y A
8 nfs1v 1812 . . . . . . . . . . . . 13 x[y / x]φ
97, 8nfan 1454 . . . . . . . . . . . 12 x(y A [y / x]φ)
10 eleq1 2097 . . . . . . . . . . . . 13 (x = y → (x Ay A))
11 sbequ12 1651 . . . . . . . . . . . . 13 (x = y → (φ ↔ [y / x]φ))
1210, 11anbi12d 442 . . . . . . . . . . . 12 (x = y → ((x A φ) ↔ (y A [y / x]φ)))
139, 12sbie 1671 . . . . . . . . . . 11 ([y / x](x A φ) ↔ (y A [y / x]φ))
1413anbi2i 430 . . . . . . . . . 10 (((x A φ) [y / x](x A φ)) ↔ ((x A φ) (y A [y / x]φ)))
15 an4 520 . . . . . . . . . 10 (((x A φ) (y A [y / x]φ)) ↔ ((x A y A) (φ [y / x]φ)))
1614, 15bitri 173 . . . . . . . . 9 (((x A φ) [y / x](x A φ)) ↔ ((x A y A) (φ [y / x]φ)))
1716imbi1i 227 . . . . . . . 8 ((((x A φ) [y / x](x A φ)) → x = y) ↔ (((x A y A) (φ [y / x]φ)) → x = y))
18 impexp 250 . . . . . . . 8 ((((x A y A) (φ [y / x]φ)) → x = y) ↔ ((x A y A) → ((φ [y / x]φ) → x = y)))
19 impexp 250 . . . . . . . 8 (((x A y A) → ((φ [y / x]φ) → x = y)) ↔ (x A → (y A → ((φ [y / x]φ) → x = y))))
2017, 18, 193bitri 195 . . . . . . 7 ((((x A φ) [y / x](x A φ)) → x = y) ↔ (x A → (y A → ((φ [y / x]φ) → x = y))))
2120albii 1356 . . . . . 6 (y(((x A φ) [y / x](x A φ)) → x = y) ↔ y(x A → (y A → ((φ [y / x]φ) → x = y))))
22 df-ral 2305 . . . . . . 7 (y A ((φ [y / x]φ) → x = y) ↔ y(y A → ((φ [y / x]φ) → x = y)))
2322imbi2i 215 . . . . . 6 ((x Ay A ((φ [y / x]φ) → x = y)) ↔ (x Ay(y A → ((φ [y / x]φ) → x = y))))
246, 21, 233bitr4i 201 . . . . 5 (y(((x A φ) [y / x](x A φ)) → x = y) ↔ (x Ay A ((φ [y / x]φ) → x = y)))
2524albii 1356 . . . 4 (xy(((x A φ) [y / x](x A φ)) → x = y) ↔ x(x Ay A ((φ [y / x]φ) → x = y)))
265, 25bitr4i 176 . . 3 (x A y A ((φ [y / x]φ) → x = y) ↔ xy(((x A φ) [y / x](x A φ)) → x = y))
274, 26anbi12i 433 . 2 ((x A φ x A y A ((φ [y / x]φ) → x = y)) ↔ (x(x A φ) xy(((x A φ) [y / x](x A φ)) → x = y)))
282, 3, 273bitr4i 201 1 (∃!x A φ ↔ (x A φ x A y A ((φ [y / x]φ) → x = y)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wex 1378   wcel 1390  [wsb 1642  ∃!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-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307
This theorem is referenced by: (None)
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