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Theorem reueq 3742
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 3232 . 2 (𝐵𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
2 moeq 3712 . . . 4 ∃*𝑥 𝑥 = 𝐵
3 mormo 3384 . . . 4 (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥𝐴 𝑥 = 𝐵)
42, 3ax-mp 5 . . 3 ∃*𝑥𝐴 𝑥 = 𝐵
5 reu5 3381 . . 3 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ (∃𝑥𝐴 𝑥 = 𝐵 ∧ ∃*𝑥𝐴 𝑥 = 𝐵))
64, 5mpbiran2 710 . 2 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
71, 6bitr4i 278 1 (𝐵𝐴 ↔ ∃!𝑥𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1539  wcel 2107  ∃*wmo 2537  wrex 3069  ∃!wreu 3377  ∃*wrmo 3378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-mo 2539  df-eu 2568  df-cleq 2728  df-clel 2815  df-rex 3070  df-rmo 3379  df-reu 3380
This theorem is referenced by:  icoshftf1o  13515  addsq2reu  27485  euoreqb  47126  isuspgrimlem  47879  inlinecirc02preu  48714
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