Theorem 3sstr4g 3135

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 3120	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	sylibr 133	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1331   C_ wss 3066

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 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079

This theorem is referenced by:  rabss2  3175  unss2  3242  sslin  3297  ssopab2  4192  xpss12  4641  coss1  4689  coss2  4690  cnvss  4707  rnss  4764  ssres  4840  ssres2  4841  imass1  4909  imass2  4910  imadif  5198  imain  5200  ssoprab2  5820  suppssfv  5971  suppssov1  5972  tposss  6136  ss2ixp  6598  isumsplit  11253  isumrpcl  11256