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Theorem seeq1 5650
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 4039 . . 3 (𝑅 = 𝑆𝑆𝑅)
2 sess1 5646 . . 3 (𝑆𝑅 → (𝑅 Se 𝐴𝑆 Se 𝐴))
31, 2syl 17 . 2 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
4 eqimss 4038 . . 3 (𝑅 = 𝑆𝑅𝑆)
5 sess1 5646 . . 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 1534  wss 3947   Se wse 5631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964  df-br 5149  df-se 5634
This theorem is referenced by:  oieq1  9535
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