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Theorem csbexga 4051
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 2999 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 abid2 2258 . . . . . . 7 {𝑦𝑦𝐵} = 𝐵
3 elex 2692 . . . . . . 7 (𝐵𝑊𝐵 ∈ V)
42, 3eqeltrid 2224 . . . . . 6 (𝐵𝑊 → {𝑦𝑦𝐵} ∈ V)
54alimi 1431 . . . . 5 (∀𝑥 𝐵𝑊 → ∀𝑥{𝑦𝑦𝐵} ∈ V)
6 spsbc 2915 . . . . 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 2279 . . . . 5 𝑥V
109sbcabel 2985 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝑦𝐵} ∈ V ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ V))
1110adantr 274 . . 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, 12eqeltrid 2224 1 ((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329  wcel 1480  {cab 2123  Vcvv 2681  [wsbc 2904  csb 2998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999
This theorem is referenced by:  csbexa  4052
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