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

Theorem vjust 3187
Description: Soundness justification theorem for df-v 3188. (Contributed by Rodolfo Medina, 27-Apr-2010.)
Assertion
Ref Expression
vjust {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}

Proof of Theorem vjust
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equid 1936 . . . . 5 𝑥 = 𝑥
21sbt 2418 . . . 4 [𝑧 / 𝑥]𝑥 = 𝑥
3 equid 1936 . . . . 5 𝑦 = 𝑦
43sbt 2418 . . . 4 [𝑧 / 𝑦]𝑦 = 𝑦
52, 42th 254 . . 3 ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
6 df-clab 2608 . . 3 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥)
7 df-clab 2608 . . 3 (𝑧 ∈ {𝑦𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
85, 6, 73bitr4i 292 . 2 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦𝑦 = 𝑦})
98eqriv 2618 1 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  [wsb 1877  wcel 1987  {cab 2607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-sb 1878  df-clab 2608  df-cleq 2614
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator