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

Theorem sbcimdv 3107
 Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
Hypothesis
Ref Expression
sbcimdv.1 (φ → (ψχ))
Assertion
Ref Expression
sbcimdv ((φ A V) → ([̣A / xψ → [̣A / xχ))
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   χ(x)   A(x)   V(x)

Proof of Theorem sbcimdv
StepHypRef Expression
1 sbcimdv.1 . . . . 5 (φ → (ψχ))
21alrimiv 1631 . . . 4 (φx(ψχ))
3 spsbc 3058 . . . 4 (A V → (x(ψχ) → [̣A / x]̣(ψχ)))
42, 3syl5 28 . . 3 (A V → (φ → [̣A / x]̣(ψχ)))
5 sbcimg 3087 . . 3 (A V → ([̣A / x]̣(ψχ) ↔ ([̣A / xψ → [̣A / xχ)))
64, 5sylibd 205 . 2 (A V → (φ → ([̣A / xψ → [̣A / xχ)))
76impcom 419 1 ((φ A V) → ([̣A / xψ → [̣A / xχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀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-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: (None)
 Copyright terms: Public domain W3C validator