Theorem sseqtrid 3275

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 3249	. . 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 3198

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 3204  df-ss 3211

This theorem is referenced by:  fssdm  5494  fndmdif  5748  fneqeql2  5752  fconst4m  5869  f1opw2  6224  ecss  6740  pw2f1odclem  7015  fopwdom  7017  ssenen  7032  phplem2  7034  fiintim  7116  casefun  7275  caseinj  7279  djufun  7294  djuinj  7296  nn0supp  9444  monoord2  10738  binom1dif  12038  znleval  14657  cnpnei  14933  cnntri  14938  cnntr  14939  cncnp  14944  cndis  14955  txdis1cn  14992  hmeontr  15027  hmeoimaf1o  15028  dvcoapbr  15421  vtxdfifiun  16103