Theorem sseqtr4d 3086

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

Hypotheses

Ref	Expression
sseqtr4d.1	|- ( ph -> A C_ B )
sseqtr4d.2	|- ( ph -> C = B )

Assertion

Ref	Expression
sseqtr4d	|- ( ph -> A C_ C )

Proof of Theorem sseqtr4d

Step	Hyp	Ref	Expression
1		sseqtr4d.1	. 2 |- ( ph -> A C_ B )
2		sseqtr4d.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2105	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3085	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1299   C_ wss 3021

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 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-in 3027  df-ss 3034

This theorem is referenced by:  syl5sseqr  3098  fnfvima  5584  tfrlemiubacc  6157  tfr1onlemubacc  6173  tfrcllemubacc  6186  rdgivallem  6208  nnnninf  6935  dfphi2  11688  ctinf  11735  toponss  11975  ssntr  12073  iscnp3  12153  cnprcl2k  12156  tgcn  12158  tgcnp  12159  ssidcn  12160  cncnp  12180  txcnp  12221  imasnopn  12249  blssec  12366  blssopn  12413  metcnp  12436  nnsf  12783  nninfsellemsuc  12792