Theorem 3sstr4i 3267

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 3254	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	mpbir 146	1 |- C C_ D

Syntax hints:   = wceq 1397   C_ wss 3199

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-in 3205  df-ss 3212

This theorem is referenced by:  undif2ss  3569  pwsnss  3888  iinuniss  4054  brab2a  4781  relopabiv  4855  rncoss  5005  imassrn  5089  rnin  5148  inimass  5155  imadiflem  5411  imainlem  5413  ssoprab2i  6115  npsspw  7696  axresscn  8085  mpomulf  8174