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Theorem seeq1 4061
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 2992 . . 3 (𝑅 = 𝑆𝑆𝑅)
2 sess1 4059 . . 3 (𝑆𝑅 → (𝑅 Se A𝑆 Se A))
31, 2syl 14 . 2 (𝑅 = 𝑆 → (𝑅 Se A𝑆 Se A))
4 eqimss 2991 . . 3 (𝑅 = 𝑆𝑅𝑆)
5 sess1 4059 . . 3 (𝑅𝑆 → (𝑆 Se A𝑅 Se A))
64, 5syl 14 . 2 (𝑅 = 𝑆 → (𝑆 Se A𝑅 Se A))
73, 6impbid 120 1 (𝑅 = 𝑆 → (𝑅 Se A𝑆 Se A))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wss 2911   Se wse 4055
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-br 3756  df-se 4056
This theorem is referenced by: (None)
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