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Theorem dfss1 3169
Description: A frequently-used variant of subclass definition df-ss 2959. (Contributed by NM, 10-Jan-2015.)
Assertion
Ref Expression
dfss1 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)

Proof of Theorem dfss1
StepHypRef Expression
1 df-ss 2959 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 incom 3157 . . 3 (𝐴𝐵) = (𝐵𝐴)
32eqeq1i 2063 . 2 ((𝐴𝐵) = 𝐴 ↔ (𝐵𝐴) = 𝐴)
41, 3bitri 177 1 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  cin 2944  wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959
This theorem is referenced by:  dfss5  3170  sseqin2  3184  onintexmid  4325  xpimasn  4797  fndmdif  5300
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