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Theorem elimhyp3v 3712
 Description: Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
Hypotheses
Ref Expression
elimhyp3v.1 (A = if(φ, A, D) → (φχ))
elimhyp3v.2 (B = if(φ, B, R) → (χθ))
elimhyp3v.3 (C = if(φ, C, S) → (θτ))
elimhyp3v.4 (D = if(φ, A, D) → (ηζ))
elimhyp3v.5 (R = if(φ, B, R) → (ζσ))
elimhyp3v.6 (S = if(φ, C, S) → (στ))
elimhyp3v.7 η
Assertion
Ref Expression
elimhyp3v τ

Proof of Theorem elimhyp3v
StepHypRef Expression
1 iftrue 3668 . . . . . 6 (φ → if(φ, A, D) = A)
21eqcomd 2358 . . . . 5 (φA = if(φ, A, D))
3 elimhyp3v.1 . . . . 5 (A = if(φ, A, D) → (φχ))
42, 3syl 15 . . . 4 (φ → (φχ))
5 iftrue 3668 . . . . . 6 (φ → if(φ, B, R) = B)
65eqcomd 2358 . . . . 5 (φB = if(φ, B, R))
7 elimhyp3v.2 . . . . 5 (B = if(φ, B, R) → (χθ))
86, 7syl 15 . . . 4 (φ → (χθ))
9 iftrue 3668 . . . . . 6 (φ → if(φ, C, S) = C)
109eqcomd 2358 . . . . 5 (φC = if(φ, C, S))
11 elimhyp3v.3 . . . . 5 (C = if(φ, C, S) → (θτ))
1210, 11syl 15 . . . 4 (φ → (θτ))
134, 8, 123bitrd 270 . . 3 (φ → (φτ))
1413ibi 232 . 2 (φτ)
15 elimhyp3v.7 . . 3 η
16 iffalse 3669 . . . . . 6 φ → if(φ, A, D) = D)
1716eqcomd 2358 . . . . 5 φD = if(φ, A, D))
18 elimhyp3v.4 . . . . 5 (D = if(φ, A, D) → (ηζ))
1917, 18syl 15 . . . 4 φ → (ηζ))
20 iffalse 3669 . . . . . 6 φ → if(φ, B, R) = R)
2120eqcomd 2358 . . . . 5 φR = if(φ, B, R))
22 elimhyp3v.5 . . . . 5 (R = if(φ, B, R) → (ζσ))
2321, 22syl 15 . . . 4 φ → (ζσ))
24 iffalse 3669 . . . . . 6 φ → if(φ, C, S) = S)
2524eqcomd 2358 . . . . 5 φS = if(φ, C, S))
26 elimhyp3v.6 . . . . 5 (S = if(φ, C, S) → (στ))
2725, 26syl 15 . . . 4 φ → (στ))
2819, 23, 273bitrd 270 . . 3 φ → (ητ))
2915, 28mpbii 202 . 2 φτ)
3014, 29pm2.61i 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-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)
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