Theorem 3sstr4g 3291

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 3276	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	sylibr 134	1 |- ( ph -> C C_ D )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is used by:  rabss2  3331  unss2  3400  sslin  3457  ssopab2  4420  xpss12  4884  coss1  4937  coss2  4938  cnvss  4955  rnss  5014  ssres  5091  ssres2  5092  imass1  5165  imass2  5166  imadif  5464  imain  5466  ssoprab2  6147  suppssov1  6302  ressuppss  6498  tposss  6521  ss2ixp  6997  isumsplit  12283  isumrpcl  12286