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Theorem pwjust 3511
 Description: Soundness justification theorem for df-pw 3512. (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 3120 . . 3 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21cbvabv 2264 . 2 {𝑥𝑥𝐴} = {𝑧𝑧𝐴}
3 sseq1 3120 . . 3 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
43cbvabv 2264 . 2 {𝑧𝑧𝐴} = {𝑦𝑦𝐴}
52, 4eqtri 2160 1 {𝑥𝑥𝐴} = {𝑦𝑦𝐴}
 Colors of variables: wff set class Syntax hints:   = wceq 1331  {cab 2125   ⊆ wss 3071 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084 This theorem is referenced by: (None)
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