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Theorem soeq2 4080
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 4078 . . . 4 (𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
2 soss 4078 . . . 4 (𝐵𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐵))
31, 2anim12i 325 . . 3 ((𝐴𝐵𝐵𝐴) → ((𝑅 Or 𝐵𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴𝑅 Or 𝐵)))
4 eqss 2987 . . 3 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 dfbi2 374 . . 3 ((𝑅 Or 𝐵𝑅 Or 𝐴) ↔ ((𝑅 Or 𝐵𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴𝑅 Or 𝐵)))
63, 4, 53imtr4i 194 . 2 (𝐴 = 𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
76bicomd 133 1 (𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wss 2944   Or wor 4059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-in 2951  df-ss 2958  df-po 4060  df-iso 4061
This theorem is referenced by: (None)
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