 Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rmoeqALT Structured version   Visualization version   GIF version

Theorem rmoeqALT 28711
 Description: Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) Obsolete version of rmoeq 3372 as of 27-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
rmoeqALT (𝐴𝑉 → ∃*𝑥𝐵 𝑥 = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem rmoeqALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
21rgenw 2908 . . 3 𝑥𝐵 (𝑥 = 𝐴𝑥 = 𝐴)
3 eqeq2 2621 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
43imbi2d 329 . . . . 5 (𝑦 = 𝐴 → ((𝑥 = 𝐴𝑥 = 𝑦) ↔ (𝑥 = 𝐴𝑥 = 𝐴)))
54ralbidv 2969 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝐵 (𝑥 = 𝐴𝑥 = 𝑦) ↔ ∀𝑥𝐵 (𝑥 = 𝐴𝑥 = 𝐴)))
65spcegv 3267 . . 3 (𝐴𝑉 → (∀𝑥𝐵 (𝑥 = 𝐴𝑥 = 𝐴) → ∃𝑦𝑥𝐵 (𝑥 = 𝐴𝑥 = 𝑦)))
72, 6mpi 20 . 2 (𝐴𝑉 → ∃𝑦𝑥𝐵 (𝑥 = 𝐴𝑥 = 𝑦))
8 nfv 1830 . . 3 𝑦 𝑥 = 𝐴
98rmo2 3492 . 2 (∃*𝑥𝐵 𝑥 = 𝐴 ↔ ∃𝑦𝑥𝐵 (𝑥 = 𝐴𝑥 = 𝑦))
107, 9sylibr 223 1 (𝐴𝑉 → ∃*𝑥𝐵 𝑥 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃*wrmo 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rmo 2904  df-v 3175 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator