Theorem sseqtrid 3277

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 3251	. . 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 1397   C_ wss 3200

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213

This theorem is referenced by:  fssdm  5497  fndmdif  5752  fneqeql2  5756  fconst4m  5874  f1opw2  6229  ecss  6745  pw2f1odclem  7020  fopwdom  7022  ssenen  7037  phplem2  7039  fiintim  7123  casefun  7284  caseinj  7288  djufun  7303  djuinj  7305  nn0supp  9454  monoord2  10749  binom1dif  12066  znleval  14686  cnpnei  14962  cnntri  14967  cnntr  14968  cncnp  14973  cndis  14984  txdis1cn  15021  hmeontr  15056  hmeoimaf1o  15057  dvcoapbr  15450  uhgrspansubgr  16147  vtxdfifiun  16167