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  9619  monoord2  10923  binom1dif  12254  ballotfilemro  13266  znleval  14988  cnpnei  15320  cnntri  15325  cnntr  15326  cncnp  15331  cndis  15342  txdis1cn  15379  hmeontr  15414  hmeoimaf1o  15415  dvcoapbr  15808  uhgrspansubgr  16518  vtxdfifiun  16538