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Theorem csbco3g 3193
 Description: Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
Hypothesis
Ref Expression
sbcco3g.1 (x = AB = C)
Assertion
Ref Expression
csbco3g (A V[A / x][B / y]D = [C / y]D)
Distinct variable groups:   x,A   x,C   x,D
Allowed substitution hints:   A(y)   B(x,y)   C(y)   D(y)   V(x,y)

Proof of Theorem csbco3g
StepHypRef Expression
1 csbnestg 3186 . 2 (A V[A / x][B / y]D = [[A / x]B / y]D)
2 elex 2867 . . . 4 (A VA V)
3 nfcvd 2490 . . . . 5 (A V → xC)
4 sbcco3g.1 . . . . 5 (x = AB = C)
53, 4csbiegf 3176 . . . 4 (A V → [A / x]B = C)
62, 5syl 15 . . 3 (A V[A / x]B = C)
76csbeq1d 3142 . 2 (A V[[A / x]B / y]D = [C / y]D)
81, 7eqtrd 2385 1 (A V[A / x][B / y]D = [C / y]D)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2859  [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-3an 936  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:  csbco3gOLD  3194
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