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Theorem csbexa 4116
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 4115 . . 3 ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → 𝐴 / 𝑥𝐵 ∈ V)
31, 2mpan 422 . 2 (∀𝑥 𝐵 ∈ V → 𝐴 / 𝑥𝐵 ∈ V)
4 csbexa.2 . 2 𝐵 ∈ V
53, 4mpg 1444 1 𝐴 / 𝑥𝐵 ∈ V
Colors of variables: wff set class
Syntax hints:  wal 1346  wcel 2141  Vcvv 2730  csb 3049
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050
This theorem is referenced by:  dfmpo  6199
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