MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcth Structured version   Visualization version   GIF version

Theorem sbcth 3759
Description: A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcth.1 𝜑
Assertion
Ref Expression
sbcth (𝐴𝑉[𝐴 / 𝑥]𝜑)

Proof of Theorem sbcth
StepHypRef Expression
1 sbcth.1 . . 3 𝜑
21ax-gen 1798 . 2 𝑥𝜑
3 spsbc 3757 . 2 (𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
42, 3mpi 20 1 (𝐴𝑉[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wcel 2107  [wsbc 3744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-sbc 3745
This theorem is referenced by:  iota4an  6483  tfinds2  7805  wunnat  17850  wunnatOLD  17851  catcfuccl  18012  catcfucclOLD  18013  dprdval  19789  opsbc2ie  31446  bj-sbceqgALT  35398  f1omptsnlem  35836  mptsnunlem  35838  topdifinffinlem  35847  relowlpssretop  35864  cdlemk35s  39429  cdlemk39s  39431  cdlemk42  39433  frege92  42301
  Copyright terms: Public domain W3C validator