Theorem sseqtrd 3065

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

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

Assertion

Ref	Expression
sseqtrd	|- ( ph -> A C_ C )

Proof of Theorem sseqtrd

Step	Hyp	Ref	Expression
1		sseqtrd.1	. 2 |- ( ph -> A C_ B )
2		sseqtrd.2	. . 3 |- ( ph -> B = C )
3	2	sseq2d 3057	. 2 |- ( ph -> ( A C_ B <-> A C_ C ) )
4	1, 3	mpbid 146	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1290   C_ wss 3002

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3008  df-ss 3015

This theorem is referenced by:  sseqtr4d  3066  fssdmd  5189  resasplitss  5205  nnaword2  6289  erssxp  6331  phpm  6637  ioodisj  9473  tgcl  11827  basgen  11843  bastop1  11846  bastop2  11847  clsss2  11892  topssnei  11925  nninfalllemn  12201