Theorem sseqtrrid 3299

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 3287	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:  resdif  5661  fimacnv  5837  tfrlem5  6585  fsumsplit  12190  fprodsplitdc  12379  phimullem  13023  ennnfonelemss  13350  prdssca  14224  prdsbas  14225  prdsplusg  14226  prdsmulr  14227  lspssid  14786  aspssid  15069  istopon  15163  sscls  15270  mopnfss  15597  plyaddlem1  15897  plymullem1  15898  lgsquadlem2  16295