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Theorem csbco 3827
Description: Composition law for chained substitutions into a class. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker csbcow 3826 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.)
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 2872 . . . . 5 (𝑧𝑦 / 𝑥𝐵[𝑦 / 𝑥]𝑧𝐵)
32sbcbii 3755 . . . 4 ([𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵[𝐴 / 𝑦][𝑦 / 𝑥]𝑧𝐵)
4 sbcco 3720 . . . 4 ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧𝐵[𝐴 / 𝑥]𝑧𝐵)
53, 4bitri 278 . . 3 ([𝐴 / 𝑦]𝑧𝑦 / 𝑥𝐵[𝐴 / 𝑥]𝑧𝐵)
65abbii 2808 . 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 2775 1 𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2110  {cab 2714  [wsbc 3694  csb 3811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-sbc 3695  df-csb 3812
This theorem is referenced by:  sbccom2  36020
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