Theorem 3sstr4g 3198

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 3183	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	sylibr 134	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1353   C_ wss 3129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142

This theorem is referenced by:  rabss2  3238  unss2  3306  sslin  3361  ssopab2  4273  xpss12  4731  coss1  4779  coss2  4780  cnvss  4797  rnss  4854  ssres  4930  ssres2  4931  imass1  5000  imass2  5001  imadif  5293  imain  5295  ssoprab2  5926  suppssfv  6074  suppssov1  6075  tposss  6242  ss2ixp  6706  isumsplit  11490  isumrpcl  11493