Theorem 3sstr3i 3037

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 3025	. 2 |- ( A C_ B <-> C C_ D )
5	1, 4	mpbi 143	1 |- C C_ D

Syntax hints:   = wceq 1284   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by: (None)