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Theorem sseqtrid 3147
Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
sseqtrid.1 𝐵𝐴
sseqtrid.2 (𝜑𝐴 = 𝐶)
Assertion
Ref Expression
sseqtrid (𝜑𝐵𝐶)

Proof of Theorem sseqtrid
StepHypRef Expression
1 sseqtrid.2 . 2 (𝜑𝐴 = 𝐶)
2 sseqtrid.1 . 2 𝐵𝐴
3 sseq2 3121 . . 3 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
43biimpa 294 . 2 ((𝐴 = 𝐶𝐵𝐴) → 𝐵𝐶)
51, 2, 4sylancl 409 1 (𝜑𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084
This theorem is referenced by:  fssdm  5287  fndmdif  5525  fneqeql2  5529  fconst4m  5640  f1opw2  5976  ecss  6470  fopwdom  6730  ssenen  6745  phplem2  6747  fiintim  6817  casefun  6970  caseinj  6974  djufun  6989  djuinj  6991  nn0supp  9036  monoord2  10257  binom1dif  11263  cnpnei  12398  cnntri  12403  cnntr  12404  cncnp  12409  cndis  12420  txdis1cn  12457  hmeontr  12492  hmeoimaf1o  12493  dvcoapbr  12850
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