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Theorem seeq2 5660
Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
seeq2 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))

Proof of Theorem seeq2
StepHypRef Expression
1 eqimss2 4055 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 sess2 5655 . . 3 (𝐵𝐴 → (𝑅 Se 𝐴𝑅 Se 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
4 eqimss 4054 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 sess2 5655 . . 3 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
73, 6impbid 212 1 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wss 3963   Se wse 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-se 5642
This theorem is referenced by:  seeq12d  5661  oieq2  9551
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