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Theorem csbid 2962
Description: Analog of sbid 1715 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbid 𝑥 / 𝑥𝐴 = 𝐴

Proof of Theorem csbid
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 2956 . 2 𝑥 / 𝑥𝐴 = {𝑦[𝑥 / 𝑥]𝑦𝐴}
2 sbcid 2877 . . 3 ([𝑥 / 𝑥]𝑦𝐴𝑦𝐴)
32abbii 2215 . 2 {𝑦[𝑥 / 𝑥]𝑦𝐴} = {𝑦𝑦𝐴}
4 abid2 2220 . 2 {𝑦𝑦𝐴} = 𝐴
51, 3, 43eqtri 2124 1 𝑥 / 𝑥𝐴 = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1299  wcel 1448  {cab 2086  [wsbc 2862  csb 2955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-sbc 2863  df-csb 2956
This theorem is referenced by:  csbeq1a  2963  fvmpt2  5436  fsumsplitf  11016
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