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Theorem seeq1 5563
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 3979 . . 3 (𝑅 = 𝑆𝑆𝑅)
2 sess1 5559 . . 3 (𝑆𝑅 → (𝑅 Se 𝐴𝑆 Se 𝐴))
31, 2syl 17 . 2 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
4 eqimss 3978 . . 3 (𝑅 = 𝑆𝑅𝑆)
5 sess1 5559 . . 3 (𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
64, 5syl 17 . 2 (𝑅 = 𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
73, 6impbid 211 1 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wss 3888   Se wse 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-v 3433  df-in 3895  df-ss 3905  df-br 5077  df-se 5547
This theorem is referenced by:  oieq1  9269
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