Theorem sseqtrrd 3136

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

Hypotheses

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

Assertion

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

Proof of Theorem sseqtrrd

Step	Hyp	Ref	Expression
1		sseqtrrd.1	. 2 |- ( ph -> A C_ B )
2		sseqtrrd.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2145	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3135	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1331   C_ wss 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  sseqtrrid  3148  fnfvima  5652  tfrlemiubacc  6227  tfr1onlemubacc  6243  tfrcllemubacc  6256  rdgivallem  6278  nnnninf  7023  dfphi2  11896  ctinf  11943  toponss  12193  ssntr  12291  iscnp3  12372  cnprcl2k  12375  tgcn  12377  tgcnp  12378  ssidcn  12379  cncnp  12399  txcnp  12440  imasnopn  12468  hmeontr  12482  blssec  12607  blssopn  12654  xmettx  12679  metcnp  12681  nnsf  13199  nninfsellemsuc  13208