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Theorem reu2 3024
 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 1619 . . 3 y(x A φ)
21eu2 2229 . 2 (∃!x(x A φ) ↔ (x(x A φ) xy(((x A φ) [y / x](x A φ)) → x = y)))
3 df-reu 2621 . 2 (∃!x A φ∃!x(x A φ))
4 df-rex 2620 . . 3 (x A φx(x A φ))
5 df-ral 2619 . . . 4 (x A y A ((φ [y / x]φ) → x = y) ↔ x(x Ay A ((φ [y / x]φ) → x = y)))
6 19.21v 1890 . . . . . 6 (y(x A → (y A → ((φ [y / x]φ) → x = y))) ↔ (x Ay(y A → ((φ [y / x]φ) → x = y))))
7 nfv 1619 . . . . . . . . . . . . 13 x y A
8 nfs1v 2106 . . . . . . . . . . . . 13 x[y / x]φ
97, 8nfan 1824 . . . . . . . . . . . 12 x(y A [y / x]φ)
10 eleq1 2413 . . . . . . . . . . . . 13 (x = y → (x Ay A))
11 sbequ12 1919 . . . . . . . . . . . . 13 (x = y → (φ ↔ [y / x]φ))
1210, 11anbi12d 691 . . . . . . . . . . . 12 (x = y → ((x A φ) ↔ (y A [y / x]φ)))
139, 12sbie 2038 . . . . . . . . . . 11 ([y / x](x A φ) ↔ (y A [y / x]φ))
1413anbi2i 675 . . . . . . . . . 10 (((x A φ) [y / x](x A φ)) ↔ ((x A φ) (y A [y / x]φ)))
15 an4 797 . . . . . . . . . 10 (((x A φ) (y A [y / x]φ)) ↔ ((x A y A) (φ [y / x]φ)))
1614, 15bitri 240 . . . . . . . . 9 (((x A φ) [y / x](x A φ)) ↔ ((x A y A) (φ [y / x]φ)))
1716imbi1i 315 . . . . . . . 8 ((((x A φ) [y / x](x A φ)) → x = y) ↔ (((x A y A) (φ [y / x]φ)) → x = y))
18 impexp 433 . . . . . . . 8 ((((x A y A) (φ [y / x]φ)) → x = y) ↔ ((x A y A) → ((φ [y / x]φ) → x = y)))
19 impexp 433 . . . . . . . 8 (((x A y A) → ((φ [y / x]φ) → x = y)) ↔ (x A → (y A → ((φ [y / x]φ) → x = y))))
2017, 18, 193bitri 262 . . . . . . 7 ((((x A φ) [y / x](x A φ)) → x = y) ↔ (x A → (y A → ((φ [y / x]φ) → x = y))))
2120albii 1566 . . . . . 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 2619 . . . . . . 7 (y A ((φ [y / x]φ) → x = y) ↔ y(y A → ((φ [y / x]φ) → x = y)))
2322imbi2i 303 . . . . . 6 ((x Ay A ((φ [y / x]φ) → x = y)) ↔ (x Ay(y A → ((φ [y / x]φ) → x = y))))
246, 21, 233bitr4i 268 . . . . 5 (y(((x A φ) [y / x](x A φ)) → x = y) ↔ (x Ay A ((φ [y / x]φ) → x = y)))
2524albii 1566 . . . 4 (xy(((x A φ) [y / x](x A φ)) → x = y) ↔ x(x Ay A ((φ [y / x]φ) → x = y)))
265, 25bitr4i 243 . . 3 (x A y A ((φ [y / x]φ) → x = y) ↔ xy(((x A φ) [y / x](x A φ)) → x = y))
274, 26anbi12i 678 . 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 268 1 (∃!x A φ ↔ (x A φ x A y A ((φ [y / x]φ) → x = y)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!weu 2204  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-ral 2619  df-rex 2620  df-reu 2621 This theorem is referenced by: (None)
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