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Theorem seeq1 5115
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 3691 . . 3 (𝑅 = 𝑆𝑆𝑅)
2 sess1 5111 . . 3 (𝑆𝑅 → (𝑅 Se 𝐴𝑆 Se 𝐴))
31, 2syl 17 . 2 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
4 eqimss 3690 . . 3 (𝑅 = 𝑆𝑅𝑆)
5 sess1 5111 . . 3 (𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
64, 5syl 17 . 2 (𝑅 = 𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
73, 6impbid 202 1 (𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wss 3607   Se wse 5100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-in 3614  df-ss 3621  df-br 4686  df-se 5103
This theorem is referenced by:  oieq1  8458
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