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Theorem csbexga 3876
 Description: The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
Assertion
Ref Expression
csbexga ((A 𝑉 x B 𝑊) → A / xB V)

Proof of Theorem csbexga
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-csb 2847 . 2 A / xB = {y[A / x]y B}
2 abid2 2155 . . . . . . 7 {yy B} = B
3 elex 2560 . . . . . . 7 (B 𝑊B V)
42, 3syl5eqel 2121 . . . . . 6 (B 𝑊 → {yy B} V)
54alimi 1341 . . . . 5 (x B 𝑊x{yy B} V)
6 spsbc 2769 . . . . 5 (A 𝑉 → (x{yy B} V → [A / x]{yy B} V))
75, 6syl5 28 . . . 4 (A 𝑉 → (x B 𝑊[A / x]{yy B} V))
87imp 115 . . 3 ((A 𝑉 x B 𝑊) → [A / x]{yy B} V)
9 nfcv 2175 . . . . 5 xV
109sbcabel 2833 . . . 4 (A 𝑉 → ([A / x]{yy B} V ↔ {y[A / x]y B} V))
1110adantr 261 . . 3 ((A 𝑉 x B 𝑊) → ([A / x]{yy B} V ↔ {y[A / x]y B} V))
128, 11mpbid 135 . 2 ((A 𝑉 x B 𝑊) → {y[A / x]y B} V)
131, 12syl5eqel 2121 1 ((A 𝑉 x B 𝑊) → A / xB V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   ∈ wcel 1390  {cab 2023  Vcvv 2551  [wsbc 2758  ⦋csb 2846 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847 This theorem is referenced by:  csbexa  3877
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