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Theorem elimhyp4v 4099
Description: Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4089). (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 4042 . . . . . . 7 (𝜑 → if(𝜑, 𝐴, 𝐷) = 𝐴)
21eqcomd 2616 . . . . . 6 (𝜑𝐴 = if(𝜑, 𝐴, 𝐷))
3 elimhyp4v.1 . . . . . 6 (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑𝜒))
42, 3syl 17 . . . . 5 (𝜑 → (𝜑𝜒))
5 iftrue 4042 . . . . . . 7 (𝜑 → if(𝜑, 𝐵, 𝑅) = 𝐵)
65eqcomd 2616 . . . . . 6 (𝜑𝐵 = if(𝜑, 𝐵, 𝑅))
7 elimhyp4v.2 . . . . . 6 (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))
86, 7syl 17 . . . . 5 (𝜑 → (𝜒𝜃))
94, 8bitrd 267 . . . 4 (𝜑 → (𝜑𝜃))
10 iftrue 4042 . . . . . 6 (𝜑 → if(𝜑, 𝐶, 𝑆) = 𝐶)
1110eqcomd 2616 . . . . 5 (𝜑𝐶 = if(𝜑, 𝐶, 𝑆))
12 elimhyp4v.3 . . . . 5 (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))
1311, 12syl 17 . . . 4 (𝜑 → (𝜃𝜏))
14 iftrue 4042 . . . . . 6 (𝜑 → if(𝜑, 𝐹, 𝐺) = 𝐹)
1514eqcomd 2616 . . . . 5 (𝜑𝐹 = if(𝜑, 𝐹, 𝐺))
16 elimhyp4v.4 . . . . 5 (𝐹 = if(𝜑, 𝐹, 𝐺) → (𝜏𝜓))
1715, 16syl 17 . . . 4 (𝜑 → (𝜏𝜓))
189, 13, 173bitrd 293 . . 3 (𝜑 → (𝜑𝜓))
1918ibi 255 . 2 (𝜑𝜓)
20 elimhyp4v.9 . . 3 𝜂
21 iffalse 4045 . . . . . . 7 𝜑 → if(𝜑, 𝐴, 𝐷) = 𝐷)
2221eqcomd 2616 . . . . . 6 𝜑𝐷 = if(𝜑, 𝐴, 𝐷))
23 elimhyp4v.5 . . . . . 6 (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))
2422, 23syl 17 . . . . 5 𝜑 → (𝜂𝜁))
25 iffalse 4045 . . . . . . 7 𝜑 → if(𝜑, 𝐵, 𝑅) = 𝑅)
2625eqcomd 2616 . . . . . 6 𝜑𝑅 = if(𝜑, 𝐵, 𝑅))
27 elimhyp4v.6 . . . . . 6 (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))
2826, 27syl 17 . . . . 5 𝜑 → (𝜁𝜎))
2924, 28bitrd 267 . . . 4 𝜑 → (𝜂𝜎))
30 iffalse 4045 . . . . . 6 𝜑 → if(𝜑, 𝐶, 𝑆) = 𝑆)
3130eqcomd 2616 . . . . 5 𝜑𝑆 = if(𝜑, 𝐶, 𝑆))
32 elimhyp4v.7 . . . . 5 (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜌))
3331, 32syl 17 . . . 4 𝜑 → (𝜎𝜌))
34 iffalse 4045 . . . . . 6 𝜑 → if(𝜑, 𝐹, 𝐺) = 𝐺)
3534eqcomd 2616 . . . . 5 𝜑𝐺 = if(𝜑, 𝐹, 𝐺))
36 elimhyp4v.8 . . . . 5 (𝐺 = if(𝜑, 𝐹, 𝐺) → (𝜌𝜓))
3735, 36syl 17 . . . 4 𝜑 → (𝜌𝜓))
3829, 33, 373bitrd 293 . . 3 𝜑 → (𝜂𝜓))
3920, 38mpbii 222 . 2 𝜑𝜓)
4019, 39pm2.61i 175 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195   = wceq 1475  ifcif 4036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037
This theorem is referenced by: (None)
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