Theorem sseqtrid 3278

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 3252	. . 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 3201

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 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214

This theorem is referenced by:  fssdm  5504  fndmdif  5761  fneqeql2  5765  fconst4m  5882  f1opw2  6239  fsuppeq  6425  fsuppeqg  6426  ecss  6788  pw2f1odclem  7063  fopwdom  7065  ssenen  7080  phplem2  7082  fiintim  7166  casefun  7327  caseinj  7331  djufun  7346  djuinj  7348  nn0supp  9498  monoord2  10794  binom1dif  12111  znleval  14732  cnpnei  15013  cnntri  15018  cnntr  15019  cncnp  15024  cndis  15035  txdis1cn  15072  hmeontr  15107  hmeoimaf1o  15108  dvcoapbr  15501  uhgrspansubgr  16201  vtxdfifiun  16221