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Theorem reu5 2824
 Description: Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
reu5 (∃!x A φ ↔ (x A φ ∃*x A φ))

Proof of Theorem reu5
StepHypRef Expression
1 eu5 2242 . 2 (∃!x(x A φ) ↔ (x(x A φ) ∃*x(x A φ)))
2 df-reu 2621 . 2 (∃!x A φ∃!x(x A φ))
3 df-rex 2620 . . 3 (x A φx(x A φ))
4 df-rmo 2622 . . 3 (∃*x A φ∃*x(x A φ))
53, 4anbi12i 678 . 2 ((x A φ ∃*x A φ) ↔ (x(x A φ) ∃*x(x A φ)))
61, 2, 53bitr4i 268 1 (∃!x A φ ↔ (x A φ ∃*x A φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∃wrex 2615  ∃!wreu 2616  ∃*wrmo 2617 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 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-rex 2620  df-reu 2621  df-rmo 2622 This theorem is referenced by:  reurex  2825  reurmo  2826  reu4  3030  reueq  3033  fncnv  5158
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