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Theorem csbco 2940
Description: Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbco 𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
Distinct variable group:   𝑦,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem csbco
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-csb 2932 . . . . . 6 𝑦 / 𝑥𝐵 = {𝑧[𝑦 / 𝑥]𝑧𝐵}
21abeq2i 2198 . . . . 5 (𝑧𝑦 / 𝑥𝐵[𝑦 / 𝑥]𝑧𝐵)
32sbcbii 2896 . . . 4 ([𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵[𝐴 / 𝑦][𝑦 / 𝑥]𝑧𝐵)
4 sbcco 2859 . . . 4 ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧𝐵[𝐴 / 𝑥]𝑧𝐵)
53, 4bitri 182 . . 3 ([𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵[𝐴 / 𝑥]𝑧𝐵)
65abbii 2203 . 2 {𝑧[𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵} = {𝑧[𝐴 / 𝑥]𝑧𝐵}
7 df-csb 2932 . 2 𝐴 / 𝑦𝑦 / 𝑥𝐵 = {𝑧[𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵}
8 df-csb 2932 . 2 𝐴 / 𝑥𝐵 = {𝑧[𝐴 / 𝑥]𝑧𝐵}
96, 7, 83eqtr4i 2118 1 𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  {cab 2074  [wsbc 2838  csb 2931
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2839  df-csb 2932
This theorem is referenced by:  csbvarg  2956  csbnest1g  2981  zisum  10738  fisum  10742  fsumsplitf  10765
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