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Theorem dfss1 3141
 Description: A frequently-used variant of subclass definition df-ss 2931. (Contributed by NM, 10-Jan-2015.)
Assertion
Ref Expression
dfss1 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)

Proof of Theorem dfss1
StepHypRef Expression
1 df-ss 2931 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 incom 3129 . . 3 (𝐴𝐵) = (𝐵𝐴)
32eqeq1i 2047 . 2 ((𝐴𝐵) = 𝐴 ↔ (𝐵𝐴) = 𝐴)
41, 3bitri 173 1 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   ∩ cin 2916   ⊆ wss 2917 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931 This theorem is referenced by:  dfss5  3142  sseqin2  3156  onintexmid  4297  xpimasn  4769  fndmdif  5272
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