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Theorem sbcth 2923
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 1426 . 2 𝑥𝜑
3 spsbc 2921 . 2 (𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
42, 3mpi 15 1 (𝐴𝑉[𝐴 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1330  wcel 1481  [wsbc 2910
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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2689  df-sbc 2911
This theorem is referenced by:  rabrsndc  3595  iota4an  5111
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