New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  sbcgf GIF version

Theorem sbcgf 3109
 Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypothesis
Ref Expression
sbcgf.1 xφ
Assertion
Ref Expression
sbcgf (A V → ([̣A / xφφ))

Proof of Theorem sbcgf
StepHypRef Expression
1 sbcgf.1 . 2 xφ
2 sbctt 3108 . 2 ((A V xφ) → ([̣A / xφφ))
31, 2mpan2 652 1 (A V → ([̣A / xφφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   ∈ wcel 1710  [̣wsbc 3046 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-or 359  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-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by:  sbc19.21g  3110  sbcg  3111  sbcabel  3123
 Copyright terms: Public domain W3C validator