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Theorem caovord3d 5613
 Description: Ordering law. (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))
caovord3d.5 (φ𝐷 𝑆)
Assertion
Ref Expression
caovord3d (φ → ((A𝐹B) = (𝐶𝐹𝐷) → (A𝑅𝐶𝐷𝑅B)))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   x,𝐷,y,z   φ,x,y,z   x,𝐹,y,z   x,𝑅,y,z   x,𝑆,y,z

Proof of Theorem caovord3d
StepHypRef Expression
1 breq1 3758 . 2 ((A𝐹B) = (𝐶𝐹𝐷) → ((A𝐹B)𝑅(𝐶𝐹B) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹B)))
2 caovordg.1 . . . 4 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝑅y ↔ (z𝐹x)𝑅(z𝐹y)))
3 caovordd.2 . . . 4 (φA 𝑆)
4 caovordd.4 . . . 4 (φ𝐶 𝑆)
5 caovordd.3 . . . 4 (φB 𝑆)
6 caovord2d.com . . . 4 ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))
72, 3, 4, 5, 6caovord2d 5612 . . 3 (φ → (A𝑅𝐶 ↔ (A𝐹B)𝑅(𝐶𝐹B)))
8 caovord3d.5 . . . 4 (φ𝐷 𝑆)
92, 8, 5, 4caovordd 5611 . . 3 (φ → (𝐷𝑅B ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹B)))
107, 9bibi12d 224 . 2 (φ → ((A𝑅𝐶𝐷𝑅B) ↔ ((A𝐹B)𝑅(𝐶𝐹B) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹B))))
111, 10syl5ibr 145 1 (φ → ((A𝐹B) = (𝐶𝐹𝐷) → (A𝑅𝐶𝐷𝑅B)))
 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:  ordpipqqs  6358  ltsrprg  6675
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