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

Theorem vjust 3438
 Description: Soundness justification theorem for df-v 3439. (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 1996 . . . . 5 𝑥 = 𝑥
21sbt 2044 . . . 4 [𝑧 / 𝑥]𝑥 = 𝑥
3 equid 1996 . . . . 5 𝑦 = 𝑦
43sbt 2044 . . . 4 [𝑧 / 𝑦]𝑦 = 𝑦
52, 42th 265 . . 3 ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
6 df-clab 2776 . . 3 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥)
7 df-clab 2776 . . 3 (𝑧 ∈ {𝑦𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
85, 6, 73bitr4i 304 . 2 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦𝑦 = 𝑦})
98eqriv 2792 1 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1522  [wsb 2042   ∈ wcel 2081  {cab 2775 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-9 2091  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-sb 2043  df-clab 2776  df-cleq 2788 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator