Theorem 3sstr4 2103

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 (_ B
3sstr4.2	|- C = A
3sstr4.3	|- D = B

Assertion

Ref	Expression
3sstr4	|- C (_ D

Proof of Theorem 3sstr4

Step	Hyp	Ref	Expression
1		3sstr4.1	. 2 |- A (_ B
2		3sstr4.2	. . 3 |- C = A
3		3sstr4.3	. . 3 |- D = B
4	2, 3	sseq12i 2090	. 2 |- (C (_ D <-> A (_ B)
5	1, 4	mpbir 190	1 |- C (_ D

Syntax hints:   = wceq 958   (_ wss 2050

This theorem is referenced by:  dmcoss 3369  rncoss 3370  imassrn 3421  rnin 3464  ssoprab2i 4014  rankval4 4712  npex 5103  axresscn 5280  cncnplem1 7771  bcthlem12 8007  ipasslem7 8492  ledir 9455  lejdir 9457  sshhococ 9464  inposet 10477  0alg 10660

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056

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