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Theorem csbvarg 3097
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 2760 . 2 (𝐴𝑉𝐴 ∈ V)
2 vex 2752 . . . . . 6 𝑦 ∈ V
3 df-csb 3070 . . . . . . 7 𝑦 / 𝑥𝑥 = {𝑧[𝑦 / 𝑥]𝑧𝑥}
4 sbcel2gv 3038 . . . . . . . 8 (𝑦 ∈ V → ([𝑦 / 𝑥]𝑧𝑥𝑧𝑦))
54abbi1dv 2307 . . . . . . 7 (𝑦 ∈ V → {𝑧[𝑦 / 𝑥]𝑧𝑥} = 𝑦)
63, 5eqtrid 2232 . . . . . 6 (𝑦 ∈ V → 𝑦 / 𝑥𝑥 = 𝑦)
72, 6ax-mp 5 . . . . 5 𝑦 / 𝑥𝑥 = 𝑦
87csbeq2i 3096 . . . 4 𝐴 / 𝑦𝑦 / 𝑥𝑥 = 𝐴 / 𝑦𝑦
9 csbco 3079 . . . 4 𝐴 / 𝑦𝑦 / 𝑥𝑥 = 𝐴 / 𝑥𝑥
10 df-csb 3070 . . . 4 𝐴 / 𝑦𝑦 = {𝑧[𝐴 / 𝑦]𝑧𝑦}
118, 9, 103eqtr3i 2216 . . 3 𝐴 / 𝑥𝑥 = {𝑧[𝐴 / 𝑦]𝑧𝑦}
12 sbcel2gv 3038 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑦]𝑧𝑦𝑧𝐴))
1312abbi1dv 2307 . . 3 (𝐴 ∈ V → {𝑧[𝐴 / 𝑦]𝑧𝑦} = 𝐴)
1411, 13eqtrid 2232 . 2 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
151, 14syl 14 1 (𝐴𝑉𝐴 / 𝑥𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wcel 2158  {cab 2173  Vcvv 2749  [wsbc 2974  csb 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975  df-csb 3070
This theorem is referenced by:  sbccsb2g  3099  csbfvg  5566  f1od2  6249  bj-sels  14937
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