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Theorem pwjust 3979
Description: Soundness justification theorem for df-pw 3980. (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 3475 . . 3 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
21cbvabv 2629 . 2 {𝑥𝑥𝐴} = {𝑧𝑧𝐴}
3 sseq1 3475 . . 3 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
43cbvabv 2629 . 2 {𝑧𝑧𝐴} = {𝑦𝑦𝐴}
52, 4eqtri 2527 1 {𝑥𝑥𝐴} = {𝑦𝑦𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1468  {cab 2491  wss 3426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-clab 2492  df-cleq 2498  df-clel 2501  df-in 3433  df-ss 3440
This theorem is referenced by: (None)
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