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Theorem csbcow 3870
Description: Composition law for chained substitutions into a class. Version of csbco 3871 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
Assertion
Ref Expression
csbcow 𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
Distinct variable groups:   𝑥,𝑦   𝑦,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem csbcow
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3856 . . . . . 6 𝑦 / 𝑥𝐵 = {𝑧[𝑦 / 𝑥]𝑧𝐵}
21abeq2i 2949 . . . . 5 (𝑧𝑦 / 𝑥𝐵[𝑦 / 𝑥]𝑧𝐵)
32sbcbii 3803 . . . 4 ([𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵[𝐴 / 𝑦][𝑦 / 𝑥]𝑧𝐵)
4 sbccow 3770 . . . 4 ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧𝐵[𝐴 / 𝑥]𝑧𝐵)
53, 4bitri 278 . . 3 ([𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵[𝐴 / 𝑥]𝑧𝐵)
65abbii 2887 . 2 {𝑧[𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵} = {𝑧[𝐴 / 𝑥]𝑧𝐵}
7 df-csb 3856 . 2 𝐴 / 𝑦𝑦 / 𝑥𝐵 = {𝑧[𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵}
8 df-csb 3856 . 2 𝐴 / 𝑥𝐵 = {𝑧[𝐴 / 𝑥]𝑧𝐵}
96, 7, 83eqtr4i 2855 1 𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2114  {cab 2800  [wsbc 3747  csb 3855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-sbc 3748  df-csb 3856
This theorem is referenced by:  csbnest1g  4353  csbvarg  4355  fvmpocurryd  7924  zsum  15066  fsum  15068  fsumsplitf  15089  zprod  15282  fprod  15286  gsumply1eq  20932  f1od2  30467  bj-csbsn  34306  disjinfi  41758  climinf2mpt  42295  climinfmpt  42296  dvmptmulf  42518  dvmptfprod  42526
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