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Theorem elimhyp2v 3711
 Description: Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
Hypotheses
Ref Expression
elimhyp2v.1 (A = if(φ, A, C) → (φχ))
elimhyp2v.2 (B = if(φ, B, D) → (χθ))
elimhyp2v.3 (C = if(φ, A, C) → (τη))
elimhyp2v.4 (D = if(φ, B, D) → (ηθ))
elimhyp2v.5 τ
Assertion
Ref Expression
elimhyp2v θ

Proof of Theorem elimhyp2v
StepHypRef Expression
1 iftrue 3668 . . . . . 6 (φ → if(φ, A, C) = A)
21eqcomd 2358 . . . . 5 (φA = if(φ, A, C))
3 elimhyp2v.1 . . . . 5 (A = if(φ, A, C) → (φχ))
42, 3syl 15 . . . 4 (φ → (φχ))
5 iftrue 3668 . . . . . 6 (φ → if(φ, B, D) = B)
65eqcomd 2358 . . . . 5 (φB = if(φ, B, D))
7 elimhyp2v.2 . . . . 5 (B = if(φ, B, D) → (χθ))
86, 7syl 15 . . . 4 (φ → (χθ))
94, 8bitrd 244 . . 3 (φ → (φθ))
109ibi 232 . 2 (φθ)
11 elimhyp2v.5 . . 3 τ
12 iffalse 3669 . . . . . 6 φ → if(φ, A, C) = C)
1312eqcomd 2358 . . . . 5 φC = if(φ, A, C))
14 elimhyp2v.3 . . . . 5 (C = if(φ, A, C) → (τη))
1513, 14syl 15 . . . 4 φ → (τη))
16 iffalse 3669 . . . . . 6 φ → if(φ, B, D) = D)
1716eqcomd 2358 . . . . 5 φD = if(φ, B, D))
18 elimhyp2v.4 . . . . 5 (D = if(φ, B, D) → (ηθ))
1917, 18syl 15 . . . 4 φ → (ηθ))
2015, 19bitrd 244 . . 3 φ → (τθ))
2111, 20mpbii 202 . 2 φθ)
2210, 21pm2.61i 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-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)
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