Theorem 3sstr4i 2984

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
3sstr4.1	|- A C_ B
3sstr4.2	|- C = A
3sstr4.3	|- D = B

Assertion

Ref	Expression
3sstr4i	|- C C_ D

Proof of Theorem 3sstr4i

Step	Hyp	Ref	Expression
1		3sstr4.1	. 2 |- A C_ B
2		3sstr4.2	. . 3 |- C = A
3		3sstr4.3	. . 3 |- D = B
4	2, 3	sseq12i 2971	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	mpbir 134	1 |- C C_ D

Syntax hints:   = wceq 1243   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  undif2ss  3299  pwsnss  3574  iinuniss  3737  brab2a  4393  rncoss  4602  imassrn  4679  rnin  4733  inimass  4740  imadiflem  4978  imainlem  4980  ssoprab2i  5593  npsspw  6569  axresscn  6936