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Theorem csbexga 3935
Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbexga ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)

Proof of Theorem csbexga
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 2922 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 abid2 2205 . . . . . . 7 {𝑦𝑦𝐵} = 𝐵
3 elex 2623 . . . . . . 7 (𝐵𝑊𝐵 ∈ V)
42, 3syl5eqel 2171 . . . . . 6 (𝐵𝑊 → {𝑦𝑦𝐵} ∈ V)
54alimi 1387 . . . . 5 (∀𝑥 𝐵𝑊 → ∀𝑥{𝑦𝑦𝐵} ∈ V)
6 spsbc 2839 . . . . 5 (𝐴𝑉 → (∀𝑥{𝑦𝑦𝐵} ∈ V → [𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
75, 6syl5 32 . . . 4 (𝐴𝑉 → (∀𝑥 𝐵𝑊[𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
87imp 122 . . 3 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → [𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V)
9 nfcv 2225 . . . . 5 𝑥V
109sbcabel 2908 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
1110adantr 270 . . 3 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
128, 11mpbid 145 . 2 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V)
131, 12syl5eqel 2171 1 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1285  wcel 1436  {cab 2071  Vcvv 2614  [wsbc 2828  csb 2921
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 2616  df-sbc 2829  df-csb 2922
This theorem is referenced by:  csbexa  3936
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