Theorem 3sstr4i 3183

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 3170	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	mpbir 145	1 |- C C_ D

Syntax hints:   = wceq 1343   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  undif2ss  3484  pwsnss  3783  iinuniss  3948  brab2a  4657  rncoss  4874  imassrn  4957  rnin  5013  inimass  5020  imadiflem  5267  imainlem  5269  ssoprab2i  5931  npsspw  7412  axresscn  7801