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Theorem vtoclibr 3213
Description: Variable to class conversion of transitive, irreflexive relation.
Hypotheses
Ref Expression
vtoclr.1 |- Rel R
vtoclr.2 |- ((xRy /\ yRz) -> xRz)
vtoclibr.3 |- -. xRx
Assertion
Ref Expression
vtoclibr |- ((ARB /\ BRC) -> ARC)
Distinct variable groups:   x,y,A   y,B   x,z,C,y   x,R,y,z   x,B
Proof of Theorem vtoclibr
StepHypRef Expression
1 breq1 2622 . . . . . . . . 9 |- (x = B -> (xRx <-> BRx))
2 breq2 2623 . . . . . . . . 9 |- (x = B -> (BRx <-> BRB))
31, 2bitrd 528 . . . . . . . 8 |- (x = B -> (xRx <-> BRB))
43negbid 611 . . . . . . 7 |- (x = B -> (-. xRx <-> -. BRB))
5 vtoclibr.3 . . . . . . 7 |- -. xRx
64, 5vtoclg 1847 . . . . . 6 |- (B e. V -> -. BRB)
7 vtoclr.1 . . . . . . . 8 |- Rel R
87brrelexi 3208 . . . . . . 7 |- (BRB -> B e. V)
98con3i 98 . . . . . 6 |- (-. B e. V -> -. BRB)
106, 9pm2.61i 126 . . . . 5 |- -. BRB
11 brprc 2661 . . . . 5 |- (-. C e. V -> (BRC <-> BRB))
1210, 11mtbiri 717 . . . 4 |- (-. C e. V -> -. BRC)
1312a3i 74 . . 3 |- (BRC -> C e. V)
14 vtoclr.2 . . . 4 |- ((xRy /\ yRz) -> xRz)
157, 14vtoclr 3211 . . 3 |- (C e. V -> ((ARB /\ BRC) -> ARC))
1613, 15syl 10 . 2 |- (BRC -> ((ARB /\ BRC) -> ARC))
1716anabsi7 497 1 |- ((ARB /\ BRC) -> ARC)
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   class class class wbr 2619  Rel wrel 3175
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185
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