Theorem 3sstr3d 3245

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 3232	. 2 |- ( ph -> ( A C_ B <-> C C_ D ) )
5	1, 4	mpbid 147	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3174

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187

This theorem is referenced by:  fnsnsplitss  5806