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Theorem sbcth 3788
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 1789 . 2 𝑥𝜑
3 spsbc 3786 . 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 1531  wcel 2098  [wsbc 3773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-sbc 3774
This theorem is referenced by:  iota4an  6531  tfinds2  7869  wunnat  17949  wunnatOLD  17950  catcfuccl  18111  catcfucclOLD  18112  dprdval  19972  opsbc2ie  32352  bj-sbceqgALT  36511  f1omptsnlem  36946  mptsnunlem  36948  topdifinffinlem  36957  relowlpssretop  36974  cdlemk35s  40540  cdlemk39s  40542  cdlemk42  40544  frege92  43527
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