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Theorem snjust 3421
Description: Soundness justification theorem for df-sn 3422. (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 2089 . . 3 (𝑥 = 𝑧 → (𝑥 = 𝐴𝑧 = 𝐴))
21cbvabv 2206 . 2 {𝑥𝑥 = 𝐴} = {𝑧𝑧 = 𝐴}
3 eqeq1 2089 . . 3 (𝑧 = 𝑦 → (𝑧 = 𝐴𝑦 = 𝐴))
43cbvabv 2206 . 2 {𝑧𝑧 = 𝐴} = {𝑦𝑦 = 𝐴}
52, 4eqtri 2103 1 {𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1285  {cab 2069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076
This theorem is referenced by: (None)
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