Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  reueq Structured version   Visualization version   GIF version

Theorem reueq 3640
 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 3213 . 2 (𝐵𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
2 moeq 3612 . . . 4 ∃*𝑥 𝑥 = 𝐵
3 mormo 3369 . . . 4 (∃*𝑥 𝑥 = 𝐵 → ∃*𝑥𝐴 𝑥 = 𝐵)
42, 3ax-mp 5 . . 3 ∃*𝑥𝐴 𝑥 = 𝐵
5 reu5 3370 . . 3 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ (∃𝑥𝐴 𝑥 = 𝐵 ∧ ∃*𝑥𝐴 𝑥 = 𝐵))
64, 5mpbiran2 697 . 2 (∃!𝑥𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝐴 𝑥 = 𝐵)
71, 6bitr4i 270 1 (𝐵𝐴 ↔ ∃!𝑥𝐴 𝑥 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1507   ∈ wcel 2050  ∃*wmo 2545  ∃wrex 3089  ∃!wreu 3090  ∃*wrmo 3091 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-cleq 2771  df-clel 2846  df-rex 3094  df-reu 3095  df-rmo 3096 This theorem is referenced by:  icoshftf1o  12676  addsq2reu  25718  euoreqb  42720  inlinecirc02preu  44149
 Copyright terms: Public domain W3C validator