Theorem sseqtrid 3288

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 3262	. . 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 3211

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 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3217  df-ss 3224

This theorem is referenced by:  fssdm  5524  fndmdif  5783  fneqeql2  5787  fconst4m  5904  f1opw2  6261  fsuppeq  6447  fsuppeqg  6448  ecss  6810  pw2f1odclem  7087  fopwdom  7089  ssenen  7105  phplem2  7107  fiintim  7191  casefun  7376  caseinj  7380  djufun  7395  djuinj  7397  nn0supp  9552  monoord2  10848  binom1dif  12173  znleval  14801  cnpnei  15084  cnntri  15089  cnntr  15090  cncnp  15095  cndis  15106  txdis1cn  15143  hmeontr  15178  hmeoimaf1o  15179  dvcoapbr  15572  uhgrspansubgr  16272  vtxdfifiun  16292