Theorem sseqtrid 3298

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 3272	. . 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 417	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   = wceq 1402   C_ wss 3220

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  fssdm  5544  fndmdif  5805  fneqeql2  5809  fconst4m  5926  f1opw2  6286  fsuppeq  6477  fsuppeqg  6478  ecss  6840  pw2f1odclem  7124  fopwdom  7126  ssenen  7142  phplem2  7144  fiintim  7228  casefun  7415  caseinj  7419  djufun  7434  djuinj  7436  nn0supp  9598  monoord2  10901  binom1dif  12232  ballotfilemro  13244  znleval  14960  cnpnei  15243  cnntri  15248  cnntr  15249  cncnp  15254  cndis  15265  txdis1cn  15302  hmeontr  15337  hmeoimaf1o  15338  dvcoapbr  15731  uhgrspansubgr  16432  vtxdfifiun  16452