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Theorem dfss1 3227
Description: A frequently-used variant of subclass definition df-ss 3034. (Contributed by NM, 10-Jan-2015.)
Assertion
Ref Expression
dfss1 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)

Proof of Theorem dfss1
StepHypRef Expression
1 df-ss 3034 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 incom 3215 . . 3 (𝐴𝐵) = (𝐵𝐴)
32eqeq1i 2107 . 2 ((𝐴𝐵) = 𝐴 ↔ (𝐵𝐴) = 𝐴)
41, 3bitri 183 1 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1299  cin 3020  wss 3021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034
This theorem is referenced by:  dfss5  3228  sseqin2  3242  onintexmid  4425  xpimasn  4923  fndmdif  5457  infiexmid  6700  ssfidc  6751  isumss  10999  znnen  11703
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