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Theorem reueq 3565
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 3209 . 2 (𝐵𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
2 moeq 3535 . . . 4 ∃*𝑥 𝑥 = 𝐵
3 mormo 3306 . . . 4 (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥𝐴 𝑥 = 𝐵)
42, 3ax-mp 5 . . 3 ∃*𝑥𝐴 𝑥 = 𝐵
5 reu5 3307 . . 3 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ (∃𝑥𝐴 𝑥 = 𝐵 ∧ ∃*𝑥𝐴 𝑥 = 𝐵))
64, 5mpbiran2 701 . 2 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
71, 6bitr4i 269 1 (𝐵𝐴 ↔ ∃!𝑥𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1652  wcel 2155  ∃*wmo 2563  wrex 3056  ∃!wreu 3057  ∃*wrmo 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-cleq 2758  df-clel 2761  df-rex 3061  df-reu 3062  df-rmo 3063
This theorem is referenced by:  icoshftf1o  12500
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