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Theorem elimhyp4v 3713
 Description: Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3703). (Contributed by NM, 16-Apr-2005.)
Hypotheses
Ref Expression
elimhyp4v.1 (A = if(φ, A, D) → (φχ))
elimhyp4v.2 (B = if(φ, B, R) → (χθ))
elimhyp4v.3 (C = if(φ, C, S) → (θτ))
elimhyp4v.4 (F = if(φ, F, G) → (τψ))
elimhyp4v.5 (D = if(φ, A, D) → (ηζ))
elimhyp4v.6 (R = if(φ, B, R) → (ζσ))
elimhyp4v.7 (S = if(φ, C, S) → (σρ))
elimhyp4v.8 (G = if(φ, F, G) → (ρψ))
elimhyp4v.9 η
Assertion
Ref Expression
elimhyp4v ψ

Proof of Theorem elimhyp4v
StepHypRef Expression
1 iftrue 3668 . . . . . . 7 (φ → if(φ, A, D) = A)
21eqcomd 2358 . . . . . 6 (φA = if(φ, A, D))
3 elimhyp4v.1 . . . . . 6 (A = if(φ, A, D) → (φχ))
42, 3syl 15 . . . . 5 (φ → (φχ))
5 iftrue 3668 . . . . . . 7 (φ → if(φ, B, R) = B)
65eqcomd 2358 . . . . . 6 (φB = if(φ, B, R))
7 elimhyp4v.2 . . . . . 6 (B = if(φ, B, R) → (χθ))
86, 7syl 15 . . . . 5 (φ → (χθ))
94, 8bitrd 244 . . . 4 (φ → (φθ))
10 iftrue 3668 . . . . . 6 (φ → if(φ, C, S) = C)
1110eqcomd 2358 . . . . 5 (φC = if(φ, C, S))
12 elimhyp4v.3 . . . . 5 (C = if(φ, C, S) → (θτ))
1311, 12syl 15 . . . 4 (φ → (θτ))
14 iftrue 3668 . . . . . 6 (φ → if(φ, F, G) = F)
1514eqcomd 2358 . . . . 5 (φF = if(φ, F, G))
16 elimhyp4v.4 . . . . 5 (F = if(φ, F, G) → (τψ))
1715, 16syl 15 . . . 4 (φ → (τψ))
189, 13, 173bitrd 270 . . 3 (φ → (φψ))
1918ibi 232 . 2 (φψ)
20 elimhyp4v.9 . . 3 η
21 iffalse 3669 . . . . . . 7 φ → if(φ, A, D) = D)
2221eqcomd 2358 . . . . . 6 φD = if(φ, A, D))
23 elimhyp4v.5 . . . . . 6 (D = if(φ, A, D) → (ηζ))
2422, 23syl 15 . . . . 5 φ → (ηζ))
25 iffalse 3669 . . . . . . 7 φ → if(φ, B, R) = R)
2625eqcomd 2358 . . . . . 6 φR = if(φ, B, R))
27 elimhyp4v.6 . . . . . 6 (R = if(φ, B, R) → (ζσ))
2826, 27syl 15 . . . . 5 φ → (ζσ))
2924, 28bitrd 244 . . . 4 φ → (ησ))
30 iffalse 3669 . . . . . 6 φ → if(φ, C, S) = S)
3130eqcomd 2358 . . . . 5 φS = if(φ, C, S))
32 elimhyp4v.7 . . . . 5 (S = if(φ, C, S) → (σρ))
3331, 32syl 15 . . . 4 φ → (σρ))
34 iffalse 3669 . . . . . 6 φ → if(φ, F, G) = G)
3534eqcomd 2358 . . . . 5 φG = if(φ, F, G))
36 elimhyp4v.8 . . . . 5 (G = if(φ, F, G) → (ρψ))
3735, 36syl 15 . . . 4 φ → (ρψ))
3829, 33, 373bitrd 270 . . 3 φ → (ηψ))
3920, 38mpbii 202 . 2 φψ)
4019, 39pm2.61i 156 1 ψ
 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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