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Theorem reueq 3650
Description: Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
Assertion
Ref Expression
reueq (𝐵𝐴 ↔ ∃!𝑥𝐴 𝑥 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem reueq
StepHypRef Expression
1 risset 3186 . 2 (𝐵𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
2 moeq 3620 . . . 4 ∃*𝑥 𝑥 = 𝐵
3 mormo 3336 . . . 4 (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥𝐴 𝑥 = 𝐵)
42, 3ax-mp 5 . . 3 ∃*𝑥𝐴 𝑥 = 𝐵
5 reu5 3337 . . 3 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ (∃𝑥𝐴 𝑥 = 𝐵 ∧ ∃*𝑥𝐴 𝑥 = 𝐵))
64, 5mpbiran2 710 . 2 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
71, 6bitr4i 281 1 (𝐵𝐴 ↔ ∃!𝑥𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1543  wcel 2110  ∃*wmo 2537  wrex 3062  ∃!wreu 3063  ∃*wrmo 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-mo 2539  df-eu 2568  df-cleq 2729  df-clel 2816  df-rex 3067  df-reu 3068  df-rmo 3069
This theorem is referenced by:  icoshftf1o  13062  addsq2reu  26321  euoreqb  44273  inlinecirc02preu  45807
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