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Theorem snjust 4396
 Description: Soundness justification theorem for df-sn 4398. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
snjust {𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem snjust
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2781 . . 3 (𝑥 = 𝑧 → (𝑥 = 𝐴𝑧 = 𝐴))
21cbvabv 2913 . 2 {𝑥𝑥 = 𝐴} = {𝑧𝑧 = 𝐴}
3 eqeq1 2781 . . 3 (𝑧 = 𝑦 → (𝑧 = 𝐴𝑦 = 𝐴))
43cbvabv 2913 . 2 {𝑧𝑧 = 𝐴} = {𝑦𝑦 = 𝐴}
52, 4eqtri 2801 1 {𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601  {cab 2762 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 2054  ax-9 2115  ax-10 2134  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769 This theorem is referenced by: (None)
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