New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  rspc3v GIF version

Theorem rspc3v 2964
 Description: 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
Hypotheses
Ref Expression
rspc3v.1 (x = A → (φχ))
rspc3v.2 (y = B → (χθ))
rspc3v.3 (z = C → (θψ))
Assertion
Ref Expression
rspc3v ((A R B S C T) → (x R y S z T φψ))
Distinct variable groups:   ψ,z   χ,x   θ,y   x,y,z,A   y,B,z   z,C   x,R   x,S,y   x,T,y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y)   χ(y,z)   θ(x,z)   B(x)   C(x,y)   R(y,z)   S(z)

Proof of Theorem rspc3v
StepHypRef Expression
1 rspc3v.1 . . . . 5 (x = A → (φχ))
21ralbidv 2634 . . . 4 (x = A → (z T φz T χ))
3 rspc3v.2 . . . . 5 (y = B → (χθ))
43ralbidv 2634 . . . 4 (y = B → (z T χz T θ))
52, 4rspc2v 2961 . . 3 ((A R B S) → (x R y S z T φz T θ))
6 rspc3v.3 . . . 4 (z = C → (θψ))
76rspcv 2951 . . 3 (C T → (z T θψ))
85, 7sylan9 638 . 2 (((A R B S) C T) → (x R y S z T φψ))
983impa 1146 1 ((A R B S C T) → (x R y S z T φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614 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-ral 2619  df-v 2861 This theorem is referenced by:  caovassg  5626  caovdig  5632  caovdirg  5633  trd  5921
 Copyright terms: Public domain W3C validator