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Theorem seeq1 5010
 Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
seeq1 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))

Proof of Theorem seeq1
StepHypRef Expression
1 eqimss2 3621 . . 3 (𝑅 = 𝑆𝑆𝑅)
2 sess1 5006 . . 3 (𝑆𝑅 → (𝑅 Se 𝐴𝑆 Se 𝐴))
31, 2syl 17 . 2 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
4 eqimss 3620 . . 3 (𝑅 = 𝑆𝑅𝑆)
5 sess1 5006 . . 3 (𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
64, 5syl 17 . 2 (𝑅 = 𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
73, 6impbid 201 1 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ⊆ wss 3540   Se wse 4995 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  ax-sep 4709 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-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-br 4584  df-se 4998 This theorem is referenced by:  oieq1  8300
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