ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caovord2d Structured version   GIF version

Theorem caovord2d 5612
Description: Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovordg.1 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝑅y ↔ (z𝐹x)𝑅(z𝐹y)))
caovordd.2 (φA 𝑆)
caovordd.3 (φB 𝑆)
caovordd.4 (φ𝐶 𝑆)
caovord2d.com ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))
Assertion
Ref Expression
caovord2d (φ → (A𝑅B ↔ (A𝐹𝐶)𝑅(B𝐹𝐶)))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   φ,x,y,z   x,𝐹,y,z   x,𝑅,y,z   x,𝑆,y,z

Proof of Theorem caovord2d
StepHypRef Expression
1 caovordg.1 . . 3 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝑅y ↔ (z𝐹x)𝑅(z𝐹y)))
2 caovordd.2 . . 3 (φA 𝑆)
3 caovordd.3 . . 3 (φB 𝑆)
4 caovordd.4 . . 3 (φ𝐶 𝑆)
51, 2, 3, 4caovordd 5611 . 2 (φ → (A𝑅B ↔ (𝐶𝐹A)𝑅(𝐶𝐹B)))
6 caovord2d.com . . . 4 ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))
76, 4, 2caovcomd 5599 . . 3 (φ → (𝐶𝐹A) = (A𝐹𝐶))
86, 4, 3caovcomd 5599 . . 3 (φ → (𝐶𝐹B) = (B𝐹𝐶))
97, 8breq12d 3768 . 2 (φ → ((𝐶𝐹A)𝑅(𝐶𝐹B) ↔ (A𝐹𝐶)𝑅(B𝐹𝐶)))
105, 9bitrd 177 1 (φ → (A𝑅B ↔ (A𝐹𝐶)𝑅(B𝐹𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390   class class class wbr 3755  (class class class)co 5455
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  caovord3d  5613  genplt2i  6492  addnqprllem  6509  addnqprulem  6510  mulnqprl  6548  mulnqpru  6549  distrlem4prl  6559  distrlem4pru  6560  1idprl  6565  1idpru  6566  ltexprlemdisj  6579  ltexprlemloc  6580  ltexprlemfl  6582  ltexprlemfu  6584  recexprlem1ssl  6604  recexprlem1ssu  6605  aptiprleml  6610  aptiprlemu  6611  cauappcvgprlemcan  6615  cauappcvgprlemlol  6618  cauappcvgprlemloc  6623  cauappcvgprlemladdfu  6625  cauappcvgprlemladdru  6627  cauappcvgprlemladdrl  6628  cauappcvgprlem1  6630  lttrsr  6670  ltsosr  6672
  Copyright terms: Public domain W3C validator