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Theorem pocl 4006
Description: Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
pocl (𝑅 Po A → ((B A 𝐶 A 𝐷 A) → (¬ B𝑅B ((B𝑅𝐶 𝐶𝑅𝐷) → B𝑅𝐷))))

Proof of Theorem pocl
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . . 7 (x = Bx = B)
21, 1breq12d 3743 . . . . . 6 (x = B → (x𝑅xB𝑅B))
32notbid 576 . . . . 5 (x = B → (¬ x𝑅x ↔ ¬ B𝑅B))
4 breq1 3733 . . . . . . 7 (x = B → (x𝑅yB𝑅y))
54anbi1d 438 . . . . . 6 (x = B → ((x𝑅y y𝑅z) ↔ (B𝑅y y𝑅z)))
6 breq1 3733 . . . . . 6 (x = B → (x𝑅zB𝑅z))
75, 6imbi12d 223 . . . . 5 (x = B → (((x𝑅y y𝑅z) → x𝑅z) ↔ ((B𝑅y y𝑅z) → B𝑅z)))
83, 7anbi12d 442 . . . 4 (x = B → ((¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z)) ↔ (¬ B𝑅B ((B𝑅y y𝑅z) → B𝑅z))))
98imbi2d 219 . . 3 (x = B → ((𝑅 Po A → (¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z))) ↔ (𝑅 Po A → (¬ B𝑅B ((B𝑅y y𝑅z) → B𝑅z)))))
10 breq2 3734 . . . . . . 7 (y = 𝐶 → (B𝑅yB𝑅𝐶))
11 breq1 3733 . . . . . . 7 (y = 𝐶 → (y𝑅z𝐶𝑅z))
1210, 11anbi12d 442 . . . . . 6 (y = 𝐶 → ((B𝑅y y𝑅z) ↔ (B𝑅𝐶 𝐶𝑅z)))
1312imbi1d 220 . . . . 5 (y = 𝐶 → (((B𝑅y y𝑅z) → B𝑅z) ↔ ((B𝑅𝐶 𝐶𝑅z) → B𝑅z)))
1413anbi2d 437 . . . 4 (y = 𝐶 → ((¬ B𝑅B ((B𝑅y y𝑅z) → B𝑅z)) ↔ (¬ B𝑅B ((B𝑅𝐶 𝐶𝑅z) → B𝑅z))))
1514imbi2d 219 . . 3 (y = 𝐶 → ((𝑅 Po A → (¬ B𝑅B ((B𝑅y y𝑅z) → B𝑅z))) ↔ (𝑅 Po A → (¬ B𝑅B ((B𝑅𝐶 𝐶𝑅z) → B𝑅z)))))
16 breq2 3734 . . . . . . 7 (z = 𝐷 → (𝐶𝑅z𝐶𝑅𝐷))
1716anbi2d 437 . . . . . 6 (z = 𝐷 → ((B𝑅𝐶 𝐶𝑅z) ↔ (B𝑅𝐶 𝐶𝑅𝐷)))
18 breq2 3734 . . . . . 6 (z = 𝐷 → (B𝑅zB𝑅𝐷))
1917, 18imbi12d 223 . . . . 5 (z = 𝐷 → (((B𝑅𝐶 𝐶𝑅z) → B𝑅z) ↔ ((B𝑅𝐶 𝐶𝑅𝐷) → B𝑅𝐷)))
2019anbi2d 437 . . . 4 (z = 𝐷 → ((¬ B𝑅B ((B𝑅𝐶 𝐶𝑅z) → B𝑅z)) ↔ (¬ B𝑅B ((B𝑅𝐶 𝐶𝑅𝐷) → B𝑅𝐷))))
2120imbi2d 219 . . 3 (z = 𝐷 → ((𝑅 Po A → (¬ B𝑅B ((B𝑅𝐶 𝐶𝑅z) → B𝑅z))) ↔ (𝑅 Po A → (¬ B𝑅B ((B𝑅𝐶 𝐶𝑅𝐷) → B𝑅𝐷)))))
22 df-po 3999 . . . . . . . 8 (𝑅 Po Ax A y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z)))
23 r3al 2338 . . . . . . . 8 (x A y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z)) ↔ xyz((x A y A z A) → (¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
2422, 23bitri 173 . . . . . . 7 (𝑅 Po Axyz((x A y A z A) → (¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
2524biimpi 113 . . . . . 6 (𝑅 Po Axyz((x A y A z A) → (¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
262519.21bbi 1427 . . . . 5 (𝑅 Po Az((x A y A z A) → (¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
272619.21bi 1426 . . . 4 (𝑅 Po A → ((x A y A z A) → (¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
2827com12 27 . . 3 ((x A y A z A) → (𝑅 Po A → (¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
299, 15, 21, 28vtocl3ga 2594 . 2 ((B A 𝐶 A 𝐷 A) → (𝑅 Po A → (¬ B𝑅B ((B𝑅𝐶 𝐶𝑅𝐷) → B𝑅𝐷))))
3029com12 27 1 (𝑅 Po A → ((B A 𝐶 A 𝐷 A) → (¬ B𝑅B ((B𝑅𝐶 𝐶𝑅𝐷) → B𝑅𝐷))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   w3a 869  wal 1224   = wceq 1226   wcel 1369  wral 2278   class class class wbr 3730   Po wpo 3997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 529  ax-in2 530  ax-io 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998
This theorem depends on definitions:  df-bi 110  df-3an 871  df-tru 1229  df-nf 1326  df-sb 1622  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-ral 2283  df-v 2531  df-un 2893  df-sn 3348  df-pr 3349  df-op 3351  df-br 3731  df-po 3999
This theorem is referenced by:  poirr  4010  potr  4011
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