Theorem 3sstr4i 3198

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 3185	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	mpbir 146	1 |- C C_ D

Syntax hints:   = wceq 1353   C_ wss 3131

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144

This theorem is referenced by:  undif2ss  3500  pwsnss  3805  iinuniss  3971  brab2a  4681  rncoss  4899  imassrn  4983  rnin  5040  inimass  5047  imadiflem  5297  imainlem  5299  ssoprab2i  5967  npsspw  7473  axresscn  7862