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Theorem soeq2 4027
 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 4025 . . . 4 (AB → (𝑅 Or B𝑅 Or A))
2 soss 4025 . . . 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 2937 . . 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 1228   ⊆ wss 2894   Or wor 4006 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-in 2901  df-ss 2908  df-po 4007  df-iso 4008 This theorem is referenced by: (None)
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