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

Theorem pwjust 4108
Description: Soundness justification theorem for df-pw 4109. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
pwjust {𝑥𝑥𝐴} = {𝑦𝑦𝐴}
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem pwjust
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sseq1 3588 . . 3 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21cbvabv 2733 . 2 {𝑥𝑥𝐴} = {𝑧𝑧𝐴}
3 sseq1 3588 . . 3 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
43cbvabv 2733 . 2 {𝑧𝑧𝐴} = {𝑦𝑦𝐴}
52, 4eqtri 2631 1 {𝑥𝑥𝐴} = {𝑦𝑦𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  {cab 2595  wss 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-in 3546  df-ss 3553
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator