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Theorem seeq2 4045
 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 2975 . . 3 (A = BBA)
2 sess2 4043 . . 3 (BA → (𝑅 Se A𝑅 Se B))
31, 2syl 14 . 2 (A = B → (𝑅 Se A𝑅 Se B))
4 eqimss 2974 . . 3 (A = BAB)
5 sess2 4043 . . 3 (AB → (𝑅 Se B𝑅 Se A))
64, 5syl 14 . 2 (A = B → (𝑅 Se B𝑅 Se A))
73, 6impbid 120 1 (A = B → (𝑅 Se A𝑅 Se B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228   ⊆ wss 2894   Se wse 4038 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rab 2293  df-v 2537  df-in 2901  df-ss 2908  df-se 4039 This theorem is referenced by: (None)
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