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Theorem csbexa 3997
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 3996 . . 3 ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → 𝐴 / 𝑥𝐵 ∈ V)
31, 2mpan 418 . 2 (∀𝑥 𝐵 ∈ V → 𝐴 / 𝑥𝐵 ∈ V)
4 csbexa.2 . 2 𝐵 ∈ V
53, 4mpg 1395 1 𝐴 / 𝑥𝐵 ∈ V
Colors of variables: wff set class
Syntax hints:  wal 1297  wcel 1448  Vcvv 2641  csb 2955
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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863  df-csb 2956
This theorem is referenced by:  dfmpo  6050
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