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Theorem csbvarg 2980
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 2652 . 2 (𝐴𝑉𝐴 ∈ V)
2 vex 2644 . . . . . 6 𝑦 ∈ V
3 df-csb 2956 . . . . . . 7 𝑦 / 𝑥𝑥 = {𝑧[𝑦 / 𝑥]𝑧𝑥}
4 sbcel2gv 2924 . . . . . . . 8 (𝑦 ∈ V → ([𝑦 / 𝑥]𝑧𝑥𝑧𝑦))
54abbi1dv 2219 . . . . . . 7 (𝑦 ∈ V → {𝑧[𝑦 / 𝑥]𝑧𝑥} = 𝑦)
63, 5syl5eq 2144 . . . . . 6 (𝑦 ∈ V → 𝑦 / 𝑥𝑥 = 𝑦)
72, 6ax-mp 7 . . . . 5 𝑦 / 𝑥𝑥 = 𝑦
87csbeq2i 2979 . . . 4 𝐴 / 𝑦𝑦 / 𝑥𝑥 = 𝐴 / 𝑦𝑦
9 csbco 2964 . . . 4 𝐴 / 𝑦𝑦 / 𝑥𝑥 = 𝐴 / 𝑥𝑥
10 df-csb 2956 . . . 4 𝐴 / 𝑦𝑦 = {𝑧[𝐴 / 𝑦]𝑧𝑦}
118, 9, 103eqtr3i 2128 . . 3 𝐴 / 𝑥𝑥 = {𝑧[𝐴 / 𝑦]𝑧𝑦}
12 sbcel2gv 2924 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑧𝑦𝑧𝐴))
1312abbi1dv 2219 . . 3 (𝐴 ∈ V → {𝑧[𝐴 / 𝑦]𝑧𝑦} = 𝐴)
1411, 13syl5eq 2144 . 2 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
151, 14syl 14 1 (𝐴𝑉𝐴 / 𝑥𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wcel 1448  {cab 2086  Vcvv 2641  [wsbc 2862  csb 2955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863  df-csb 2956
This theorem is referenced by:  sbccsb2g  2982  csbfvg  5391  f1od2  6062  bj-sels  12693
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