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Theorem reueq 3729
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 3229 . 2 (𝐵𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
2 moeq 3699 . . . 4 ∃*𝑥 𝑥 = 𝐵
3 mormo 3380 . . . 4 (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥𝐴 𝑥 = 𝐵)
42, 3ax-mp 5 . . 3 ∃*𝑥𝐴 𝑥 = 𝐵
5 reu5 3377 . . 3 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ (∃𝑥𝐴 𝑥 = 𝐵 ∧ ∃*𝑥𝐴 𝑥 = 𝐵))
64, 5mpbiran2 708 . 2 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
71, 6bitr4i 277 1 (𝐵𝐴 ↔ ∃!𝑥𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wcel 2106  ∃*wmo 2531  wrex 3069  ∃!wreu 3373  ∃*wrmo 3374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-mo 2533  df-eu 2562  df-cleq 2723  df-clel 2809  df-rex 3070  df-rmo 3375  df-reu 3376
This theorem is referenced by:  icoshftf1o  13433  addsq2reu  26870  euoreqb  45589  inlinecirc02preu  47122
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