Theorem sseqtrid 3217

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
sseqtrid.1	|- B C_ A
sseqtrid.2	|- ( ph -> A = C )

Assertion

Ref	Expression
sseqtrid	|- ( ph -> B C_ C )

Proof of Theorem sseqtrid

Step	Hyp	Ref	Expression
1		sseqtrid.2	. 2 |- ( ph -> A = C )
2		sseqtrid.1	. 2 |- B C_ A
3		sseq2 3191	. . 3 |- ( A = C -> ( B C_ A <-> B C_ C ) )
4	3	biimpa 296	. 2 |- ( ( A = C /\ B C_ A ) -> B C_ C )
5	1, 2, 4	sylancl 413	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   = wceq 1363   C_ wss 3141

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-in 3147  df-ss 3154

This theorem is referenced by:  fssdm  5392  fndmdif  5634  fneqeql2  5638  fconst4m  5749  f1opw2  6090  ecss  6589  fopwdom  6849  ssenen  6864  phplem2  6866  fiintim  6941  casefun  7097  caseinj  7101  djufun  7116  djuinj  7118  nn0supp  9241  monoord2  10490  binom1dif  11508  cnpnei  13990  cnntri  13995  cnntr  13996  cncnp  14001  cndis  14012  txdis1cn  14049  hmeontr  14084  hmeoimaf1o  14085  dvcoapbr  14442