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Theorem reueq 3033
 Description: Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
Assertion
Ref Expression
reueq (B A∃!x A x = B)
Distinct variable groups:   x,A   x,B

Proof of Theorem reueq
StepHypRef Expression
1 risset 2661 . 2 (B Ax A x = B)
2 moeq 3012 . . . 4 ∃*x x = B
3 mormo 2823 . . . 4 (∃*x x = B∃*x A x = B)
42, 3ax-mp 5 . . 3 ∃*x A x = B
5 reu5 2824 . . 3 (∃!x A x = B ↔ (x A x = B ∃*x A x = B))
64, 5mpbiran2 885 . 2 (∃!x A x = Bx A x = B)
71, 6bitr4i 243 1 (B A∃!x A x = B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ∃wrex 2615  ∃!wreu 2616  ∃*wrmo 2617 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-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2620  df-reu 2621  df-rmo 2622  df-v 2861 This theorem is referenced by: (None)
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