Theorem 3sstr4g 3145

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr4g.1	|- ( ph -> A C_ B )
3sstr4g.2	|- C = A
3sstr4g.3	|- D = B

Assertion

Ref	Expression
3sstr4g	|- ( ph -> C C_ D )

Proof of Theorem 3sstr4g

Step	Hyp	Ref	Expression
1		3sstr4g.1	. 2 |- ( ph -> A C_ B )
2		3sstr4g.2	. . 3 |- C = A
3		3sstr4g.3	. . 3 |- D = B
4	2, 3	sseq12i 3130	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	sylibr 133	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1332   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089

This theorem is referenced by:  rabss2  3185  unss2  3252  sslin  3307  ssopab2  4205  xpss12  4654  coss1  4702  coss2  4703  cnvss  4720  rnss  4777  ssres  4853  ssres2  4854  imass1  4922  imass2  4923  imadif  5211  imain  5213  ssoprab2  5835  suppssfv  5986  suppssov1  5987  tposss  6151  ss2ixp  6613  isumsplit  11292  isumrpcl  11295