Theorem sseqtrrid 3243

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 3231	1 |- ( ph -> B C_ C )

Syntax hints:   -> wi 4   = wceq 1372   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178

This theorem is referenced by:  resdif  5543  fimacnv  5708  tfrlem5  6399  fsumsplit  11660  fprodsplitdc  11849  phimullem  12489  ennnfonelemss  12723  prdssca  13049  prdsbas  13050  prdsplusg  13051  prdsmulr  13052  lspssid  14104  istopon  14427  sscls  14534  mopnfss  14861  plyaddlem1  15161  plymullem1  15162  lgsquadlem2  15497