Theorem 3sstr4g 3040

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 3025	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	sylibr 132	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1284   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  rabss2  3077  unss2  3143  sslin  3192  ssopab2  4030  xpss12  4463  coss1  4509  coss2  4510  cnvss  4526  rnss  4582  ssres  4655  ssres2  4656  imass1  4720  imass2  4721  imadif  4999  imain  5001  ssoprab2  5581  suppssfv  5728  suppssov1  5729  tposss  5884