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Theorem soeq2 5015
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 5013 . . . 4 (𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
2 soss 5013 . . . 4 (𝐵𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐵))
31, 2anim12i 589 . . 3 ((𝐴𝐵𝐵𝐴) → ((𝑅 Or 𝐵𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴𝑅 Or 𝐵)))
4 eqss 3598 . . 3 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 dfbi2 659 . . 3 ((𝑅 Or 𝐵𝑅 Or 𝐴) ↔ ((𝑅 Or 𝐵𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴𝑅 Or 𝐵)))
63, 4, 53imtr4i 281 . 2 (𝐴 = 𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
76bicomd 213 1 (𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wss 3555   Or wor 4994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-in 3562  df-ss 3569  df-po 4995  df-so 4996
This theorem is referenced by:  weeq2  5063  wemapso2  8402  oemapso  8523  fin2i  9061  isfin2-2  9085  fin1a2lem10  9175  zorn2lem7  9268  zornn0g  9271  opsrtoslem2  19404  sltsolem1  31525  soeq12d  37085  aomclem1  37101
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