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Theorem csbexga 3881
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 2850 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 abid2 2158 . . . . . . 7 {𝑦𝑦𝐵} = 𝐵
3 elex 2563 . . . . . . 7 (𝐵𝑊𝐵 ∈ V)
42, 3syl5eqel 2124 . . . . . 6 (𝐵𝑊 → {𝑦𝑦𝐵} ∈ V)
54alimi 1344 . . . . 5 (∀𝑥 𝐵𝑊 → ∀𝑥{𝑦𝑦𝐵} ∈ V)
6 spsbc 2772 . . . . 5 (𝐴𝑉 → (∀𝑥{𝑦𝑦𝐵} ∈ V → [𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
75, 6syl5 28 . . . 4 (𝐴𝑉 → (∀𝑥 𝐵𝑊[𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V))
87imp 115 . . 3 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → [𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V)
9 nfcv 2178 . . . . 5 𝑥V
109sbcabel 2836 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
1110adantr 261 . . 3 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
128, 11mpbid 135 . 2 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V)
131, 12syl5eqel 2124 1 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wal 1241  wcel 1393  {cab 2026  Vcvv 2554  [wsbc 2761  csb 2849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-sbc 2762  df-csb 2850
This theorem is referenced by:  csbexa  3882
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