Theorem 3sstr4i 2656

Description: Substitution of equality in both sides of a subclass relationship. (The proof was 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 2643	. 2 |- (C C_ D <-> A C_ B)
5	1, 4	mpbir 207	1 |- C C_ D

Syntax hints:   = wceq 1298   C_ wss 2593

This theorem is referenced by:  dmcossOLD 4212  rncoss 4213  imassrn 4278  rnin 4326  ssoprab2i 4937  rankval4 5813  npex 6243  axresscn 6420  axresscnOLD 6421  cncnplem1 9051  bcthlem12 9288  ipasslem7 9837  lediri 11093  lejdiri 11095  sshhococi 11102  mayetes3i 11310  posprs 14581  inposet 14620  0alg 15103  strss 16711

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605

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