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  5873  f1opw2  6228  ecss  6744  pw2f1odclem  7019  fopwdom  7021  ssenen  7036  phplem2  7038  fiintim  7122  casefun  7283  caseinj  7287  djufun  7302  djuinj  7304  nn0supp  9453  monoord2  10747  binom1dif  12047  znleval  14666  cnpnei  14942  cnntri  14947  cnntr  14948  cncnp  14953  cndis  14964  txdis1cn  15001  hmeontr  15036  hmeoimaf1o  15037  dvcoapbr  15430  uhgrspansubgr  16127  vtxdfifiun  16147