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

Theorem sbciegf 3077
 Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
sbciegf.1 xψ
sbciegf.2 (x = A → (φψ))
Assertion
Ref Expression
sbciegf (A V → ([̣A / xφψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)   V(x)

Proof of Theorem sbciegf
StepHypRef Expression
1 sbciegf.1 . 2 xψ
2 sbciegf.2 . . 3 (x = A → (φψ))
32ax-gen 1546 . 2 x(x = A → (φψ))
4 sbciegft 3076 . 2 ((A V xψ x(x = A → (φψ))) → ([̣A / xφψ))
51, 3, 4mp3an23 1269 1 (A V → ([̣A / xφψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ 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-3an 936  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:  sbcieg  3078  opelopabf  4711
 Copyright terms: Public domain W3C validator