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

Theorem iseqsetv-clel 2816
Description: Alternate proof of iseqsetv-cleq 2802. The expression 𝑥𝑥 = 𝐴 does not depend on a particular choice of the set variable. Use this theorem in contexts where df-cleq 2725 or ax-ext 2704 is not referenced elsewhere in your proof. It is proven from a specific implementation (class builder, axiom df-clab 2711) of the primitive term 𝑥𝐴. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
iseqsetv-clel (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem iseqsetv-clel
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 issettru 2815 . 2 (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝑧 ∣ ⊤})
2 issettru 2815 . 2 (∃𝑦 𝑦 = 𝐴𝐴 ∈ {𝑧 ∣ ⊤})
31, 2bitr4i 278 1 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1535  wtru 1536  wex 1774  wcel 2104  {cab 2710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-clel 2812
This theorem is referenced by:  elisset  2819  bj-issettruALTV  36816  bj-issetwt  36818  bj-vtoclg1f1  36860
  Copyright terms: Public domain W3C validator