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Theorem pwjust 4501
 Description: Soundness justification theorem for df-pw 4502. (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 3942 . . 3 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21cbvabv 2866 . 2 {𝑥𝑥𝐴} = {𝑧𝑧𝐴}
3 sseq1 3942 . . 3 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
43cbvabv 2866 . 2 {𝑧𝑧𝐴} = {𝑦𝑦𝐴}
52, 4eqtri 2821 1 {𝑥𝑥𝐴} = {𝑦𝑦𝐴}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  {cab 2776   ⊆ wss 3883 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-in 3890  df-ss 3900 This theorem is referenced by: (None)
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