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Theorem soeq2 4979
 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 4977 . . . 4 (𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
2 soss 4977 . . . 4 (𝐵𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐵))
31, 2anim12i 588 . . 3 ((𝐴𝐵𝐵𝐴) → ((𝑅 Or 𝐵𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴𝑅 Or 𝐵)))
4 eqss 3583 . . 3 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 dfbi2 658 . . 3 ((𝑅 Or 𝐵𝑅 Or 𝐴) ↔ ((𝑅 Or 𝐵𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴𝑅 Or 𝐵)))
63, 4, 53imtr4i 280 . 2 (𝐴 = 𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
76bicomd 212 1 (𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ⊆ wss 3540   Or wor 4958 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-in 3547  df-ss 3554  df-po 4959  df-so 4960 This theorem is referenced by:  weeq2  5027  wemapso2  8341  oemapso  8462  fin2i  9000  isfin2-2  9024  fin1a2lem10  9114  zorn2lem7  9207  zornn0g  9210  opsrtoslem2  19306  sltsolem1  31067  soeq12d  36626  aomclem1  36642
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