Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-elisset Structured version   Visualization version   GIF version

Theorem bj-elisset 34590
 Description: Remove from elisset 2834 dependency on ax-ext 2730 (and on df-cleq 2751 and df-v 3412). This proof uses only df-clab 2737 and df-clel 2831 on top of first-order logic. It only requires ax-1--7 and sp 2181. Use bj-elissetv 34589 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-elisset (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-elisset
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bj-elissetv 34589 . 2 (𝐴𝑉 → ∃𝑦 𝑦 = 𝐴)
2 bj-denotes 34584 . 2 (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
31, 2sylib 221 1 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539  ∃wex 1782   ∈ wcel 2112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-clel 2831 This theorem is referenced by:  bj-isseti  34592  bj-ceqsalt  34600  bj-ceqsalg  34603  bj-spcimdv  34609  bj-vtoclg1f  34632
 Copyright terms: Public domain W3C validator