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Theorem elimhyp2v 4492
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 4434 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐴)
21eqcomd 2807 . . . . 5 (𝜑𝐴 = if(𝜑, 𝐴, 𝐶))
3 elimhyp2v.1 . . . . 5 (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜑𝜒))
42, 3syl 17 . . . 4 (𝜑 → (𝜑𝜒))
5 iftrue 4434 . . . . . 6 (𝜑 → if(𝜑, 𝐵, 𝐷) = 𝐵)
65eqcomd 2807 . . . . 5 (𝜑𝐵 = if(𝜑, 𝐵, 𝐷))
7 elimhyp2v.2 . . . . 5 (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))
86, 7syl 17 . . . 4 (𝜑 → (𝜒𝜃))
94, 8bitrd 282 . . 3 (𝜑 → (𝜑𝜃))
109ibi 270 . 2 (𝜑𝜃)
11 elimhyp2v.5 . . 3 𝜏
12 iffalse 4437 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐶)
1312eqcomd 2807 . . . . 5 𝜑𝐶 = if(𝜑, 𝐴, 𝐶))
14 elimhyp2v.3 . . . . 5 (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏𝜂))
1513, 14syl 17 . . . 4 𝜑 → (𝜏𝜂))
16 iffalse 4437 . . . . . 6 𝜑 → if(𝜑, 𝐵, 𝐷) = 𝐷)
1716eqcomd 2807 . . . . 5 𝜑𝐷 = if(𝜑, 𝐵, 𝐷))
18 elimhyp2v.4 . . . . 5 (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂𝜃))
1917, 18syl 17 . . . 4 𝜑 → (𝜂𝜃))
2015, 19bitrd 282 . . 3 𝜑 → (𝜏𝜃))
2111, 20mpbii 236 . 2 𝜑𝜃)
2210, 21pm2.61i 185 1 𝜃
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  ifcif 4428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-if 4429
This theorem is referenced by:  omlsi  29191
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