Theorem sseqtrid 3242

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 3216	. . 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 1372   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178

This theorem is referenced by:  fssdm  5439  fndmdif  5684  fneqeql2  5688  fconst4m  5803  f1opw2  6151  ecss  6662  pw2f1odclem  6930  fopwdom  6932  ssenen  6947  phplem2  6949  fiintim  7027  casefun  7186  caseinj  7190  djufun  7205  djuinj  7207  nn0supp  9346  monoord2  10629  binom1dif  11740  znleval  14357  cnpnei  14633  cnntri  14638  cnntr  14639  cncnp  14644  cndis  14655  txdis1cn  14692  hmeontr  14727  hmeoimaf1o  14728  dvcoapbr  15121