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Theorem sbcth 3060
 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 (A V → [̣A / xφ)

Proof of Theorem sbcth
StepHypRef Expression
1 sbcth.1 . . 3 φ
21ax-gen 1546 . 2 xφ
3 spsbc 3058 . 2 (A V → (xφ → [̣A / xφ))
42, 3mpi 16 1 (A V → [̣A / xφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 1710  [̣wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861  df-sbc 3047 This theorem is referenced by:  iota4an  4358
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