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Theorem csbexa 3933
Description: The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypotheses
Ref Expression
csbexa.1 𝐴 ∈ V
csbexa.2 𝐵 ∈ V
Assertion
Ref Expression
csbexa 𝐴 / 𝑥𝐵 ∈ V

Proof of Theorem csbexa
StepHypRef Expression
1 csbexa.1 . . 3 𝐴 ∈ V
2 csbexga 3932 . . 3 ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → 𝐴 / 𝑥𝐵 ∈ V)
31, 2mpan 415 . 2 (∀𝑥 𝐵 ∈ V → 𝐴 / 𝑥𝐵 ∈ V)
4 csbexa.2 . 2 𝐵 ∈ V
53, 4mpg 1381 1 𝐴 / 𝑥𝐵 ∈ V
Colors of variables: wff set class
Syntax hints:  wal 1283  wcel 1434  Vcvv 2612  csb 2919
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-sbc 2827  df-csb 2920
This theorem is referenced by:  dfmpt2  5921
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