Theorem 3sstr3i 3244

Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr3.1	|- A C_ B
3sstr3.2	|- A = C
3sstr3.3	|- B = D

Assertion

Ref	Expression
3sstr3i	|- C C_ D

Proof of Theorem 3sstr3i

Step	Hyp	Ref	Expression
1		3sstr3.1	. 2 |- A C_ B
2		3sstr3.2	. . 3 |- A = C
3		3sstr3.3	. . 3 |- B = D
4	2, 3	sseq12i 3232	. 2 |- ( A C_ B <-> C C_ D )
5	1, 4	mpbi 145	1 |- C C_ D

Syntax hints:   = wceq 1375   C_ wss 3177

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 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-11 1532  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-in 3183  df-ss 3190

This theorem is referenced by: (None)