MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caovord Structured version   Visualization version   GIF version

Theorem caovord 7636
Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)
Hypotheses
Ref Expression
caovord.1 𝐴 ∈ V
caovord.2 𝐵 ∈ V
caovord.3 (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
Assertion
Ref Expression
caovord (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovord
StepHypRef Expression
1 oveq1 7431 . . . 4 (𝑧 = 𝐶 → (𝑧𝐹𝐴) = (𝐶𝐹𝐴))
2 oveq1 7431 . . . 4 (𝑧 = 𝐶 → (𝑧𝐹𝐵) = (𝐶𝐹𝐵))
31, 2breq12d 5163 . . 3 (𝑧 = 𝐶 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵) ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
43bibi2d 341 . 2 (𝑧 = 𝐶 → ((𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)) ↔ (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
5 caovord.1 . . 3 𝐴 ∈ V
6 caovord.2 . . 3 𝐵 ∈ V
7 breq1 5153 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
8 oveq2 7432 . . . . . . 7 (𝑥 = 𝐴 → (𝑧𝐹𝑥) = (𝑧𝐹𝐴))
98breq1d 5160 . . . . . 6 (𝑥 = 𝐴 → ((𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)))
107, 9bibi12d 344 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦))))
11 breq2 5154 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
12 oveq2 7432 . . . . . . 7 (𝑦 = 𝐵 → (𝑧𝐹𝑦) = (𝑧𝐹𝐵))
1312breq2d 5162 . . . . . 6 (𝑦 = 𝐵 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))
1411, 13bibi12d 344 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))))
1510, 14sylan9bb 508 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))))
1615imbi2d 339 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) ↔ (𝑧𝑆 → (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))))
17 caovord.3 . . 3 (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
185, 6, 16, 17vtocl2 3552 . 2 (𝑧𝑆 → (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))
194, 18vtoclga 3563 1 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  Vcvv 3471   class class class wbr 5150  (class class class)co 7424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-iota 6503  df-fv 6559  df-ov 7427
This theorem is referenced by:  caovord2  7637  caovord3  7638  genpcl  11037
  Copyright terms: Public domain W3C validator