New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  keephyp3v GIF version

Theorem keephyp3v 3718
 Description: Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
Hypotheses
Ref Expression
keephyp3v.1 (A = if(φ, A, D) → (ρχ))
keephyp3v.2 (B = if(φ, B, R) → (χθ))
keephyp3v.3 (C = if(φ, C, S) → (θτ))
keephyp3v.4 (D = if(φ, A, D) → (ηζ))
keephyp3v.5 (R = if(φ, B, R) → (ζσ))
keephyp3v.6 (S = if(φ, C, S) → (στ))
keephyp3v.7 ρ
keephyp3v.8 η
Assertion
Ref Expression
keephyp3v τ

Proof of Theorem keephyp3v
StepHypRef Expression
1 keephyp3v.7 . . 3 ρ
2 iftrue 3668 . . . . . 6 (φ → if(φ, A, D) = A)
32eqcomd 2358 . . . . 5 (φA = if(φ, A, D))
4 keephyp3v.1 . . . . 5 (A = if(φ, A, D) → (ρχ))
53, 4syl 15 . . . 4 (φ → (ρχ))
6 iftrue 3668 . . . . . 6 (φ → if(φ, B, R) = B)
76eqcomd 2358 . . . . 5 (φB = if(φ, B, R))
8 keephyp3v.2 . . . . 5 (B = if(φ, B, R) → (χθ))
97, 8syl 15 . . . 4 (φ → (χθ))
10 iftrue 3668 . . . . . 6 (φ → if(φ, C, S) = C)
1110eqcomd 2358 . . . . 5 (φC = if(φ, C, S))
12 keephyp3v.3 . . . . 5 (C = if(φ, C, S) → (θτ))
1311, 12syl 15 . . . 4 (φ → (θτ))
145, 9, 133bitrd 270 . . 3 (φ → (ρτ))
151, 14mpbii 202 . 2 (φτ)
16 keephyp3v.8 . . 3 η
17 iffalse 3669 . . . . . 6 φ → if(φ, A, D) = D)
1817eqcomd 2358 . . . . 5 φD = if(φ, A, D))
19 keephyp3v.4 . . . . 5 (D = if(φ, A, D) → (ηζ))
2018, 19syl 15 . . . 4 φ → (ηζ))
21 iffalse 3669 . . . . . 6 φ → if(φ, B, R) = R)
2221eqcomd 2358 . . . . 5 φR = if(φ, B, R))
23 keephyp3v.5 . . . . 5 (R = if(φ, B, R) → (ζσ))
2422, 23syl 15 . . . 4 φ → (ζσ))
25 iffalse 3669 . . . . . 6 φ → if(φ, C, S) = S)
2625eqcomd 2358 . . . . 5 φS = if(φ, C, S))
27 keephyp3v.6 . . . . 5 (S = if(φ, C, S) → (στ))
2826, 27syl 15 . . . 4 φ → (στ))
2920, 24, 283bitrd 270 . . 3 φ → (ητ))
3016, 29mpbii 202 . 2 φτ)
3115, 30pm2.61i 156 1 τ
 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-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