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 )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220

This proof depends on 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 proof 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 used by:  fssdm  5549  fndmdif  5814  fneqeql2  5818  fconst4m  5935  f1opw2  6296  fsuppeq  6487  fsuppeqg  6488  ecss  6850  pw2f1odclem  7134  fopwdom  7136  ssenen  7152  phplem2  7154  fiintim  7238  casefun  7425  caseinj  7429  djufun  7444  djuinj  7446  nn0supp  9623  monoord2  10936  binom1dif  12270  ballotfilemro  13315  znleval  15037  cnpnei  15369  cnntri  15374  cnntr  15375  cncnp  15380  cndis  15391  txdis1cn  15428  hmeontr  15463  hmeoimaf1o  15464  dvcoapbr  15857  uhgrspansubgr  16616  vtxdfifiun  16636