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Theorem sseqtrid 3197
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 3171 . . 3 (𝐴 = 𝐶 → (𝐵𝐴𝐵𝐶))
43biimpa 294 . 2 ((𝐴 = 𝐶𝐵𝐴) → 𝐵𝐶)
51, 2, 4sylancl 411 1 (𝜑𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wss 3121
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134
This theorem is referenced by:  fssdm  5362  fndmdif  5601  fneqeql2  5605  fconst4m  5716  f1opw2  6055  ecss  6554  fopwdom  6814  ssenen  6829  phplem2  6831  fiintim  6906  casefun  7062  caseinj  7066  djufun  7081  djuinj  7083  nn0supp  9187  monoord2  10433  binom1dif  11450  cnpnei  13013  cnntri  13018  cnntr  13019  cncnp  13024  cndis  13035  txdis1cn  13072  hmeontr  13107  hmeoimaf1o  13108  dvcoapbr  13465
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