Theorem 3sstr3g 3266

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

Hypotheses

Ref	Expression
3sstr3g.1	|- ( ph -> A C_ B )
3sstr3g.2	|- A = C
3sstr3g.3	|- B = D

Assertion

Ref	Expression
3sstr3g	|- ( ph -> C C_ D )

Proof of Theorem 3sstr3g

Step	Hyp	Ref	Expression
1		3sstr3g.1	. 2 |- ( ph -> A C_ B )
2		3sstr3g.2	. . 3 |- A = C
3		3sstr3g.3	. . 3 |- B = D
4	2, 3	sseq12i 3252	. 2 |- ( A C_ B <-> C C_ D )
5	1, 4	sylib 122	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3197

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  hmeontr  14987