ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbexa GIF version

Theorem csbexa 4144
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 4143 . . 3 ((𝐴 ∈ V ∧ ∀𝑥 𝐵 ∈ V) → 𝐴 / 𝑥𝐵 ∈ V)
31, 2mpan 424 . 2 (∀𝑥 𝐵 ∈ V → 𝐴 / 𝑥𝐵 ∈ V)
4 csbexa.2 . 2 𝐵 ∈ V
53, 4mpg 1461 1 𝐴 / 𝑥𝐵 ∈ V
Colors of variables: wff set class
Syntax hints:  wal 1361  wcel 2158  Vcvv 2749  csb 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975  df-csb 3070
This theorem is referenced by:  dfmpo  6237
  Copyright terms: Public domain W3C validator