 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  keephyp2v GIF version

Theorem keephyp2v 3717
 Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3703). (Contributed by NM, 16-Apr-2005.)
Hypotheses
Ref Expression
keephyp2v.1 (A = if(φ, A, C) → (ψχ))
keephyp2v.2 (B = if(φ, B, D) → (χθ))
keephyp2v.3 (C = if(φ, A, C) → (τη))
keephyp2v.4 (D = if(φ, B, D) → (ηθ))
keephyp2v.5 ψ
keephyp2v.6 τ
Assertion
Ref Expression
keephyp2v θ

Proof of Theorem keephyp2v
StepHypRef Expression
1 keephyp2v.5 . . 3 ψ
2 iftrue 3668 . . . . . 6 (φ → if(φ, A, C) = A)
32eqcomd 2358 . . . . 5 (φA = if(φ, A, C))
4 keephyp2v.1 . . . . 5 (A = if(φ, A, C) → (ψχ))
53, 4syl 15 . . . 4 (φ → (ψχ))
6 iftrue 3668 . . . . . 6 (φ → if(φ, B, D) = B)
76eqcomd 2358 . . . . 5 (φB = if(φ, B, D))
8 keephyp2v.2 . . . . 5 (B = if(φ, B, D) → (χθ))
97, 8syl 15 . . . 4 (φ → (χθ))
105, 9bitrd 244 . . 3 (φ → (ψθ))
111, 10mpbii 202 . 2 (φθ)
12 keephyp2v.6 . . 3 τ
13 iffalse 3669 . . . . . 6 φ → if(φ, A, C) = C)
1413eqcomd 2358 . . . . 5 φC = if(φ, A, C))
15 keephyp2v.3 . . . . 5 (C = if(φ, A, C) → (τη))
1614, 15syl 15 . . . 4 φ → (τη))
17 iffalse 3669 . . . . . 6 φ → if(φ, B, D) = D)
1817eqcomd 2358 . . . . 5 φD = if(φ, B, D))
19 keephyp2v.4 . . . . 5 (D = if(φ, B, D) → (ηθ))
2018, 19syl 15 . . . 4 φ → (ηθ))
2116, 20bitrd 244 . . 3 φ → (τθ))
2212, 21mpbii 202 . 2 φθ)
2311, 22pm2.61i 156 1 θ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ifcif 3662 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-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