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Theorem elimhyp3v 4292
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 4236 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐷) = 𝐴)
21eqcomd 2766 . . . . 5 (𝜑𝐴 = if(𝜑, 𝐴, 𝐷))
3 elimhyp3v.1 . . . . 5 (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑𝜒))
42, 3syl 17 . . . 4 (𝜑 → (𝜑𝜒))
5 iftrue 4236 . . . . . 6 (𝜑 → if(𝜑, 𝐵, 𝑅) = 𝐵)
65eqcomd 2766 . . . . 5 (𝜑𝐵 = if(𝜑, 𝐵, 𝑅))
7 elimhyp3v.2 . . . . 5 (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))
86, 7syl 17 . . . 4 (𝜑 → (𝜒𝜃))
9 iftrue 4236 . . . . . 6 (𝜑 → if(𝜑, 𝐶, 𝑆) = 𝐶)
109eqcomd 2766 . . . . 5 (𝜑𝐶 = if(𝜑, 𝐶, 𝑆))
11 elimhyp3v.3 . . . . 5 (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))
1210, 11syl 17 . . . 4 (𝜑 → (𝜃𝜏))
134, 8, 123bitrd 294 . . 3 (𝜑 → (𝜑𝜏))
1413ibi 256 . 2 (𝜑𝜏)
15 elimhyp3v.7 . . 3 𝜂
16 iffalse 4239 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐷) = 𝐷)
1716eqcomd 2766 . . . . 5 𝜑𝐷 = if(𝜑, 𝐴, 𝐷))
18 elimhyp3v.4 . . . . 5 (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))
1917, 18syl 17 . . . 4 𝜑 → (𝜂𝜁))
20 iffalse 4239 . . . . . 6 𝜑 → if(𝜑, 𝐵, 𝑅) = 𝑅)
2120eqcomd 2766 . . . . 5 𝜑𝑅 = if(𝜑, 𝐵, 𝑅))
22 elimhyp3v.5 . . . . 5 (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))
2321, 22syl 17 . . . 4 𝜑 → (𝜁𝜎))
24 iffalse 4239 . . . . . 6 𝜑 → if(𝜑, 𝐶, 𝑆) = 𝑆)
2524eqcomd 2766 . . . . 5 𝜑𝑆 = if(𝜑, 𝐶, 𝑆))
26 elimhyp3v.6 . . . . 5 (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜏))
2725, 26syl 17 . . . 4 𝜑 → (𝜎𝜏))
2819, 23, 273bitrd 294 . . 3 𝜑 → (𝜂𝜏))
2915, 28mpbii 223 . 2 𝜑𝜏)
3014, 29pm2.61i 176 1 𝜏
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1632  ifcif 4230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-if 4231
This theorem is referenced by:  sseliALT  4943
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