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Theorem snjust 4209
Description: Soundness justification theorem for df-sn 4211. (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 2655 . . 3 (𝑥 = 𝑧 → (𝑥 = 𝐴𝑧 = 𝐴))
21cbvabv 2776 . 2 {𝑥𝑥 = 𝐴} = {𝑧𝑧 = 𝐴}
3 eqeq1 2655 . . 3 (𝑧 = 𝑦 → (𝑧 = 𝐴𝑦 = 𝐴))
43cbvabv 2776 . 2 {𝑧𝑧 = 𝐴} = {𝑦𝑦 = 𝐴}
52, 4eqtri 2673 1 {𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  {cab 2637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644
This theorem is referenced by: (None)
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