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Theorem reu3 3026
 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 2825 . . 3 (∃!x A φx A φ)
2 reu6 3025 . . . 4 (∃!x A φy A x A (φx = y))
3 bi1 178 . . . . . 6 ((φx = y) → (φx = y))
43ralimi 2689 . . . . 5 (x A (φx = y) → x A (φx = y))
54reximi 2721 . . . 4 (y A x A (φx = y) → y A x A (φx = y))
62, 5sylbi 187 . . 3 (∃!x A φy A x A (φx = y))
71, 6jca 518 . 2 (∃!x A φ → (x A φ y A x A (φx = y)))
8 rexex 2673 . . . 4 (y A x A (φx = y) → yx A (φx = y))
98anim2i 552 . . 3 ((x A φ y A x A (φx = y)) → (x A φ yx A (φx = y)))
10 nfv 1619 . . . . 5 y(x A φ)
1110eu3 2230 . . . 4 (∃!x(x A φ) ↔ (x(x A φ) yx((x A φ) → x = y)))
12 df-reu 2621 . . . 4 (∃!x A φ∃!x(x A φ))
13 df-rex 2620 . . . . 5 (x A φx(x A φ))
14 df-ral 2619 . . . . . . 7 (x A (φx = y) ↔ x(x A → (φx = y)))
15 impexp 433 . . . . . . . 8 (((x A φ) → x = y) ↔ (x A → (φx = y)))
1615albii 1566 . . . . . . 7 (x((x A φ) → x = y) ↔ x(x A → (φx = y)))
1714, 16bitr4i 243 . . . . . 6 (x A (φx = y) ↔ x((x A φ) → x = y))
1817exbii 1582 . . . . 5 (yx A (φx = y) ↔ yx((x A φ) → x = y))
1913, 18anbi12i 678 . . . 4 ((x A φ yx A (φx = y)) ↔ (x(x A φ) yx((x A φ) → x = y)))
2011, 12, 193bitr4i 268 . . 3 (∃!x A φ ↔ (x A φ yx A (φx = y)))
219, 20sylibr 203 . 2 ((x A φ y A x A (φx = y)) → ∃!x A φ)
227, 21impbii 180 1 (∃!x A φ ↔ (x A φ y A x A (φx = y)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622 This theorem is referenced by:  reu7  3031
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