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Theorem soeq2 4119
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 4117 . . . 4 (𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
2 soss 4117 . . . 4 (𝐵𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐵))
31, 2anim12i 331 . . 3 ((𝐴𝐵𝐵𝐴) → ((𝑅 Or 𝐵𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴𝑅 Or 𝐵)))
4 eqss 3029 . . 3 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 dfbi2 380 . . 3 ((𝑅 Or 𝐵𝑅 Or 𝐴) ↔ ((𝑅 Or 𝐵𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴𝑅 Or 𝐵)))
63, 4, 53imtr4i 199 . 2 (𝐴 = 𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
76bicomd 139 1 (𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wss 2988   Or wor 4098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-in 2994  df-ss 3001  df-po 4099  df-iso 4100
This theorem is referenced by: (None)
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