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Theorem csbco 3145
 Description: Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbco [A / y][y / x]B = [A / x]B
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 3137 . . . . . 6 [y / x]B = {z y / xz B}
21abeq2i 2460 . . . . 5 (z [y / x]B ↔ [̣y / xz B)
32sbcbii 3101 . . . 4 ([̣A / yz [y / x]B ↔ [̣A / y]̣[̣y / xz B)
4 sbcco 3068 . . . 4 ([̣A / y]̣[̣y / xz B ↔ [̣A / xz B)
53, 4bitri 240 . . 3 ([̣A / yz [y / x]B ↔ [̣A / xz B)
65abbii 2465 . 2 {z A / yz [y / x]B} = {z A / xz B}
7 df-csb 3137 . 2 [A / y][y / x]B = {z A / yz [y / x]B}
8 df-csb 3137 . 2 [A / x]B = {z A / xz B}
96, 7, 83eqtr4i 2383 1 [A / y][y / x]B = [A / x]B
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137 This theorem is referenced by:  csbvarg  3163  csbnest1g  3188  eqerlem  5960
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