Theorem 3sstr3d 3186

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Hypotheses

Ref	Expression
3sstr3d.1	|- ( ph -> A C_ B )
3sstr3d.2	|- ( ph -> A = C )
3sstr3d.3	|- ( ph -> B = D )

Assertion

Ref	Expression
3sstr3d	|- ( ph -> C C_ D )

Proof of Theorem 3sstr3d

Step	Hyp	Ref	Expression
1		3sstr3d.1	. 2 |- ( ph -> A C_ B )
2		3sstr3d.2	. . 3 |- ( ph -> A = C )
3		3sstr3d.3	. . 3 |- ( ph -> B = D )
4	2, 3	sseq12d 3173	. 2 |- ( ph -> ( A C_ B <-> C C_ D ) )
5	1, 4	mpbid 146	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1343   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  fnsnsplitss  5684