Theorem sseqtrrid 3276

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
sseqtrrid.1	|- B C_ A
sseqtrrid.2	|- ( ph -> C = A )

Assertion

Ref	Expression
sseqtrrid	|- ( ph -> B C_ C )

Proof of Theorem sseqtrrid

Step	Hyp	Ref	Expression
1		sseqtrrid.1	. . 3 |- B C_ A
2	1	a1i 9	. 2 |- ( ph -> B C_ A )
3		sseqtrrid.2	. 2 |- ( ph -> C = A )
4	2, 3	sseqtrrd 3264	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3198

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3204  df-ss 3211

This theorem is referenced by:  resdif  5602  fimacnv  5772  tfrlem5  6475  fsumsplit  11958  fprodsplitdc  12147  phimullem  12787  ennnfonelemss  13021  prdssca  13348  prdsbas  13349  prdsplusg  13350  prdsmulr  13351  lspssid  14404  istopon  14727  sscls  14834  mopnfss  15161  plyaddlem1  15461  plymullem1  15462  lgsquadlem2  15797