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Theorem csbvarg 2947
Description: The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbvarg (𝐴𝑉𝐴 / 𝑥𝑥 = 𝐴)

Proof of Theorem csbvarg
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2624 . 2 (𝐴𝑉𝐴 ∈ V)
2 vex 2618 . . . . . 6 𝑦 ∈ V
3 df-csb 2923 . . . . . . 7 𝑦 / 𝑥𝑥 = {𝑧[𝑦 / 𝑥]𝑧𝑥}
4 sbcel2gv 2891 . . . . . . . 8 (𝑦 ∈ V → ([𝑦 / 𝑥]𝑧𝑥𝑧𝑦))
54abbi1dv 2204 . . . . . . 7 (𝑦 ∈ V → {𝑧[𝑦 / 𝑥]𝑧𝑥} = 𝑦)
63, 5syl5eq 2129 . . . . . 6 (𝑦 ∈ V → 𝑦 / 𝑥𝑥 = 𝑦)
72, 6ax-mp 7 . . . . 5 𝑦 / 𝑥𝑥 = 𝑦
87csbeq2i 2946 . . . 4 𝐴 / 𝑦𝑦 / 𝑥𝑥 = 𝐴 / 𝑦𝑦
9 csbco 2931 . . . 4 𝐴 / 𝑦𝑦 / 𝑥𝑥 = 𝐴 / 𝑥𝑥
10 df-csb 2923 . . . 4 𝐴 / 𝑦𝑦 = {𝑧[𝐴 / 𝑦]𝑧𝑦}
118, 9, 103eqtr3i 2113 . . 3 𝐴 / 𝑥𝑥 = {𝑧[𝐴 / 𝑦]𝑧𝑦}
12 sbcel2gv 2891 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑧𝑦𝑧𝐴))
1312abbi1dv 2204 . . 3 (𝐴 ∈ V → {𝑧[𝐴 / 𝑦]𝑧𝑦} = 𝐴)
1411, 13syl5eq 2129 . 2 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
151, 14syl 14 1 (𝐴𝑉𝐴 / 𝑥𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  {cab 2071  Vcvv 2615  [wsbc 2829  csb 2922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-sbc 2830  df-csb 2923
This theorem is referenced by:  sbccsb2g  2949  csbfvg  5299  f1od2  5951  bj-sels  11235
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