Theorem sseqtrid 3274

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 3248	. . 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 1395   C_ wss 3197

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  fssdm  5488  fndmdif  5742  fneqeql2  5746  fconst4m  5863  f1opw2  6218  ecss  6731  pw2f1odclem  7003  fopwdom  7005  ssenen  7020  phplem2  7022  fiintim  7104  casefun  7263  caseinj  7267  djufun  7282  djuinj  7284  nn0supp  9432  monoord2  10720  binom1dif  12013  znleval  14632  cnpnei  14908  cnntri  14913  cnntr  14914  cncnp  14919  cndis  14930  txdis1cn  14967  hmeontr  15002  hmeoimaf1o  15003  dvcoapbr  15396  vtxdfifiun  16056