New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  dedth4v GIF version

Theorem dedth4v 3709
 Description: Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3707. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
Hypotheses
Ref Expression
dedth4v.1 (A = if(φ, A, R) → (ψχ))
dedth4v.2 (B = if(φ, B, S) → (χθ))
dedth4v.3 (C = if(φ, C, T) → (θτ))
dedth4v.4 (D = if(φ, D, U) → (τη))
dedth4v.5 η
Assertion
Ref Expression
dedth4v (φψ)

Proof of Theorem dedth4v
StepHypRef Expression
1 dedth4v.1 . . . 4 (A = if(φ, A, R) → (ψχ))
2 dedth4v.2 . . . 4 (B = if(φ, B, S) → (χθ))
3 dedth4v.3 . . . 4 (C = if(φ, C, T) → (θτ))
4 dedth4v.4 . . . 4 (D = if(φ, D, U) → (τη))
5 dedth4v.5 . . . 4 η
61, 2, 3, 4, 5dedth4h 3706 . . 3 (((φ φ) (φ φ)) → ψ)
76anidms 626 . 2 ((φ φ) → ψ)
87anidms 626 1 (φψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ifcif 3662 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-if 3663 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator