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

Theorem sbab 2475
 Description: The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable. (Contributed by NM, 14-Sep-2003.)
Assertion
Ref Expression
sbab (x = yA = {z [y / x]z A})
Distinct variable groups:   z,A   x,z   y,z
Allowed substitution hints:   A(x,y)

Proof of Theorem sbab
StepHypRef Expression
1 sbequ12 1919 . 2 (x = y → (z A ↔ [y / x]z A))
21abbi2dv 2468 1 (x = yA = {z [y / x]z A})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339 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 This theorem is referenced by:  sbcel12g  3151  sbceqg  3152
 Copyright terms: Public domain W3C validator