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Theorem csbexga 3973
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 2935 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 abid2 2209 . . . . . . 7 {𝑦𝑦𝐵} = 𝐵
3 elex 2631 . . . . . . 7 (𝐵𝑊𝐵 ∈ V)
42, 3syl5eqel 2175 . . . . . 6 (𝐵𝑊 → {𝑦𝑦𝐵} ∈ V)
54alimi 1390 . . . . 5 (∀𝑥 𝐵𝑊 → ∀𝑥{𝑦𝑦𝐵} ∈ V)
6 spsbc 2852 . . . . 5 (𝐴𝑉 → (∀𝑥{𝑦𝑦𝐵} ∈ V → [𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
75, 6syl5 32 . . . 4 (𝐴𝑉 → (∀𝑥 𝐵𝑊[𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
87imp 123 . . 3 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → [𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V)
9 nfcv 2229 . . . . 5 𝑥V
109sbcabel 2921 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
1110adantr 271 . . 3 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
128, 11mpbid 146 . 2 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V)
131, 12syl5eqel 2175 1 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1288  wcel 1439  {cab 2075  Vcvv 2620  [wsbc 2841  csb 2934
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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-sbc 2842  df-csb 2935
This theorem is referenced by:  csbexa  3974
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