Theorem sseqtrid 3292

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 3266	. . 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 1398   C_ wss 3214

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227

This theorem is referenced by:  fssdm  5529  fndmdif  5788  fneqeql2  5792  fconst4m  5909  f1opw2  6269  fsuppeq  6460  fsuppeqg  6461  ecss  6823  pw2f1odclem  7100  fopwdom  7102  ssenen  7118  phplem2  7120  fiintim  7204  casefun  7389  caseinj  7393  djufun  7408  djuinj  7410  nn0supp  9569  monoord2  10872  binom1dif  12198  ballotfilemro  13210  znleval  14927  cnpnei  15210  cnntri  15215  cnntr  15216  cncnp  15221  cndis  15232  txdis1cn  15269  hmeontr  15304  hmeoimaf1o  15305  dvcoapbr  15698  uhgrspansubgr  16398  vtxdfifiun  16418