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Theorem csbco 2886
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 2878 . . . . . 6 𝑦 / 𝑥𝐵 = {𝑧[𝑦 / 𝑥]𝑧𝐵}
21abeq2i 2162 . . . . 5 (𝑧𝑦 / 𝑥𝐵[𝑦 / 𝑥]𝑧𝐵)
32sbcbii 2842 . . . 4 ([𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵[𝐴 / 𝑦][𝑦 / 𝑥]𝑧𝐵)
4 sbcco 2805 . . . 4 ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧𝐵[𝐴 / 𝑥]𝑧𝐵)
53, 4bitri 177 . . 3 ([𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵[𝐴 / 𝑥]𝑧𝐵)
65abbii 2167 . 2 {𝑧[𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵} = {𝑧[𝐴 / 𝑥]𝑧𝐵}
7 df-csb 2878 . 2 𝐴 / 𝑦𝑦 / 𝑥𝐵 = {𝑧[𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵}
8 df-csb 2878 . 2 𝐴 / 𝑥𝐵 = {𝑧[𝐴 / 𝑥]𝑧𝐵}
96, 7, 83eqtr4i 2084 1 𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1257  wcel 1407  {cab 2040  [wsbc 2784  csb 2877
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-sbc 2785  df-csb 2878
This theorem is referenced by:  csbvarg  2902  csbnest1g  2926
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