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Theorem keephyp3v 4357
Description: Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
Hypotheses
Ref Expression
keephyp3v.1 (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜌𝜒))
keephyp3v.2 (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))
keephyp3v.3 (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))
keephyp3v.4 (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))
keephyp3v.5 (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))
keephyp3v.6 (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜏))
keephyp3v.7 𝜌
keephyp3v.8 𝜂
Assertion
Ref Expression
keephyp3v 𝜏

Proof of Theorem keephyp3v
StepHypRef Expression
1 keephyp3v.7 . . 3 𝜌
2 iftrue 4292 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐷) = 𝐴)
32eqcomd 2819 . . . . 5 (𝜑𝐴 = if(𝜑, 𝐴, 𝐷))
4 keephyp3v.1 . . . . 5 (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜌𝜒))
53, 4syl 17 . . . 4 (𝜑 → (𝜌𝜒))
6 iftrue 4292 . . . . . 6 (𝜑 → if(𝜑, 𝐵, 𝑅) = 𝐵)
76eqcomd 2819 . . . . 5 (𝜑𝐵 = if(𝜑, 𝐵, 𝑅))
8 keephyp3v.2 . . . . 5 (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))
97, 8syl 17 . . . 4 (𝜑 → (𝜒𝜃))
10 iftrue 4292 . . . . . 6 (𝜑 → if(𝜑, 𝐶, 𝑆) = 𝐶)
1110eqcomd 2819 . . . . 5 (𝜑𝐶 = if(𝜑, 𝐶, 𝑆))
12 keephyp3v.3 . . . . 5 (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))
1311, 12syl 17 . . . 4 (𝜑 → (𝜃𝜏))
145, 9, 133bitrd 296 . . 3 (𝜑 → (𝜌𝜏))
151, 14mpbii 224 . 2 (𝜑𝜏)
16 keephyp3v.8 . . 3 𝜂
17 iffalse 4295 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐷) = 𝐷)
1817eqcomd 2819 . . . . 5 𝜑𝐷 = if(𝜑, 𝐴, 𝐷))
19 keephyp3v.4 . . . . 5 (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))
2018, 19syl 17 . . . 4 𝜑 → (𝜂𝜁))
21 iffalse 4295 . . . . . 6 𝜑 → if(𝜑, 𝐵, 𝑅) = 𝑅)
2221eqcomd 2819 . . . . 5 𝜑𝑅 = if(𝜑, 𝐵, 𝑅))
23 keephyp3v.5 . . . . 5 (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))
2422, 23syl 17 . . . 4 𝜑 → (𝜁𝜎))
25 iffalse 4295 . . . . . 6 𝜑 → if(𝜑, 𝐶, 𝑆) = 𝑆)
2625eqcomd 2819 . . . . 5 𝜑𝑆 = if(𝜑, 𝐶, 𝑆))
27 keephyp3v.6 . . . . 5 (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜏))
2826, 27syl 17 . . . 4 𝜑 → (𝜎𝜏))
2920, 24, 283bitrd 296 . . 3 𝜑 → (𝜂𝜏))
3016, 29mpbii 224 . 2 𝜑𝜏)
3115, 30pm2.61i 176 1 𝜏
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197   = wceq 1637  ifcif 4286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-if 4287
This theorem is referenced by:  sseliALT  4993
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