Theorem 3sstr4g 3236

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 3221	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	sylibr 134	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  rabss2  3276  unss2  3344  sslin  3399  ssopab2  4323  xpss12  4783  coss1  4834  coss2  4835  cnvss  4852  rnss  4909  ssres  4986  ssres2  4987  imass1  5058  imass2  5059  imadif  5355  imain  5357  ssoprab2  6003  suppssfv  6156  suppssov1  6157  tposss  6334  ss2ixp  6800  isumsplit  11835  isumrpcl  11838