Theorem 3sstr4g 3283

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 3268	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	sylibr 134	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3213

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3219  df-ss 3226

This theorem is referenced by:  rabss2  3323  unss2  3392  sslin  3449  ssopab2  4396  xpss12  4859  coss1  4912  coss2  4913  cnvss  4930  rnss  4989  ssres  5066  ssres2  5067  imass1  5139  imass2  5140  imadif  5438  imain  5440  ssoprab2  6111  suppssov1  6265  ressuppss  6456  tposss  6479  ss2ixp  6948  isumsplit  12181  isumrpcl  12184