Theorem 3sstr4i 3234

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 3221	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	mpbir 146	1 |- C C_ D

Syntax hints:   = wceq 1373   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  undif2ss  3536  pwsnss  3844  iinuniss  4010  brab2a  4728  relopabiv  4801  rncoss  4949  imassrn  5033  rnin  5092  inimass  5099  imadiflem  5353  imainlem  5355  ssoprab2i  6034  npsspw  7584  axresscn  7973  mpomulf  8062