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Theorem sseqtrid 3206
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 3180 . . 3 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
43biimpa 296 . 2 ((𝐴 = 𝐶𝐵𝐴) → 𝐵𝐶)
51, 2, 4sylancl 413 1 (𝜑𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wss 3130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3136  df-ss 3143
This theorem is referenced by:  fssdm  5381  fndmdif  5622  fneqeql2  5626  fconst4m  5737  f1opw2  6077  ecss  6576  fopwdom  6836  ssenen  6851  phplem2  6853  fiintim  6928  casefun  7084  caseinj  7088  djufun  7103  djuinj  7105  nn0supp  9228  monoord2  10477  binom1dif  11495  cnpnei  13722  cnntri  13727  cnntr  13728  cncnp  13733  cndis  13744  txdis1cn  13781  hmeontr  13816  hmeoimaf1o  13817  dvcoapbr  14174
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