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Theorem snjust 4549
Description: Soundness justification theorem for df-sn 4551. (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 2828 . . 3 (𝑥 = 𝑧 → (𝑥 = 𝐴𝑧 = 𝐴))
21cbvabv 2892 . 2 {𝑥𝑥 = 𝐴} = {𝑧𝑧 = 𝐴}
3 eqeq1 2828 . . 3 (𝑧 = 𝑦 → (𝑧 = 𝐴𝑦 = 𝐴))
43cbvabv 2892 . 2 {𝑧𝑧 = 𝐴} = {𝑦𝑦 = 𝐴}
52, 4eqtri 2847 1 {𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  {cab 2802
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 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817
This theorem is referenced by: (None)
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