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Theorem csbexa 3883
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 3882 . . 3 ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → 𝐴 / 𝑥𝐵 ∈ V)
31, 2mpan 400 . 2 (∀𝑥 𝐵 ∈ V → 𝐴 / 𝑥𝐵 ∈ V)
4 csbexa.2 . 2 𝐵 ∈ V
53, 4mpg 1340 1 𝐴 / 𝑥𝐵 ∈ V
Colors of variables: wff set class
Syntax hints:  wal 1241  wcel 1393  Vcvv 2554  csb 2849
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-sbc 2762  df-csb 2850
This theorem is referenced by:  dfmpt2  5807
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