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Theorem soeq2 4044
Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
Assertion
Ref Expression
soeq2 (A = B → (𝑅 Or A𝑅 Or B))

Proof of Theorem soeq2
StepHypRef Expression
1 soss 4042 . . . 4 (AB → (𝑅 Or B𝑅 Or A))
2 soss 4042 . . . 4 (BA → (𝑅 Or A𝑅 Or B))
31, 2anim12i 321 . . 3 ((AB BA) → ((𝑅 Or B𝑅 Or A) (𝑅 Or A𝑅 Or B)))
4 eqss 2954 . . 3 (A = B ↔ (AB BA))
5 dfbi2 368 . . 3 ((𝑅 Or B𝑅 Or A) ↔ ((𝑅 Or B𝑅 Or A) (𝑅 Or A𝑅 Or B)))
63, 4, 53imtr4i 190 . 2 (A = B → (𝑅 Or B𝑅 Or A))
76bicomd 129 1 (A = B → (𝑅 Or A𝑅 Or B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wss 2911   Or wor 4023
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-in 2918  df-ss 2925  df-po 4024  df-iso 4025
This theorem is referenced by: (None)
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