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Theorem csbco 3825
 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 3812 . . . . . 6 𝑦 / 𝑥𝐵 = {𝑧[𝑦 / 𝑥]𝑧𝐵}
21abeq2i 2917 . . . . 5 (𝑧𝑦 / 𝑥𝐵[𝑦 / 𝑥]𝑧𝐵)
32sbcbii 3757 . . . 4 ([𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵[𝐴 / 𝑦][𝑦 / 𝑥]𝑧𝐵)
4 sbcco 3729 . . . 4 ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧𝐵[𝐴 / 𝑥]𝑧𝐵)
53, 4bitri 276 . . 3 ([𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵[𝐴 / 𝑥]𝑧𝐵)
65abbii 2861 . 2 {𝑧[𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵} = {𝑧[𝐴 / 𝑥]𝑧𝐵}
7 df-csb 3812 . 2 𝐴 / 𝑦𝑦 / 𝑥𝐵 = {𝑧[𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵}
8 df-csb 3812 . 2 𝐴 / 𝑥𝐵 = {𝑧[𝐴 / 𝑥]𝑧𝐵}
96, 7, 83eqtr4i 2829 1 𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1522   ∈ wcel 2081  {cab 2775  [wsbc 3706  ⦋csb 3811 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-v 3439  df-sbc 3707  df-csb 3812 This theorem is referenced by:  csbnest1g  4296  csbvarg  4298  fvmpocurryd  7788  zsum  14908  fsum  14910  fsumsplitf  14931  zprod  15124  fprod  15128  gsumply1eq  20156  f1od2  30145  bj-csbsn  33792  sbccom2  34935  disjinfi  40994  climinf2mpt  41537  climinfmpt  41538  dvmptmulf  41763  dvmptfprod  41771
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