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

Theorem sbciedf 3081
 Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
sbcied.1 (φA V)
sbcied.2 ((φ x = A) → (ψχ))
sbciedf.3 xφ
sbciedf.4 (φ → Ⅎxχ)
Assertion
Ref Expression
sbciedf (φ → ([̣A / xψχ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)   χ(x)   V(x)

Proof of Theorem sbciedf
StepHypRef Expression
1 sbcied.1 . 2 (φA V)
2 sbciedf.4 . 2 (φ → Ⅎxχ)
3 sbciedf.3 . . 3 xφ
4 sbcied.2 . . . 4 ((φ x = A) → (ψχ))
54ex 423 . . 3 (φ → (x = A → (ψχ)))
63, 5alrimi 1765 . 2 (φx(x = A → (ψχ)))
7 sbciegft 3076 . 2 ((A V xχ x(x = A → (ψχ))) → ([̣A / xψχ))
81, 2, 6, 7syl3anc 1182 1 (φ → ([̣A / xψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ 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-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:  sbcied  3082  sbc2iegf  3112  csbiebt  3172  sbcnestgf  3183
 Copyright terms: Public domain W3C validator