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Theorem pwjust 4380
 Description: Soundness justification theorem for df-pw 4381. (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 3845 . . 3 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21cbvabv 2914 . 2 {𝑥𝑥𝐴} = {𝑧𝑧𝐴}
3 sseq1 3845 . . 3 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
43cbvabv 2914 . 2 {𝑧𝑧𝐴} = {𝑦𝑦𝐴}
52, 4eqtri 2802 1 {𝑥𝑥𝐴} = {𝑦𝑦𝐴}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601  {cab 2763   ⊆ wss 3792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-in 3799  df-ss 3806 This theorem is referenced by: (None)
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