Theorem 3sstr3 2096

Description: Substitution of equality in both sides of a subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr3.1	|- A (_ B
3sstr3.2	|- A = C
3sstr3.3	|- B = D

Assertion

Ref	Expression
3sstr3	|- C (_ D

Proof of Theorem 3sstr3

Step	Hyp	Ref	Expression
1		3sstr3.1	. 2 |- A (_ B
2		3sstr3.2	. . 3 |- A = C
3		3sstr3.3	. . 3 |- B = D
4	2, 3	sseq12i 2084	. 2 |- (A (_ B <-> C (_ D)
5	1, 4	mpbi 189	1 |- C (_ D

Syntax hints:   = wceq 955   (_ wss 2044

This theorem is referenced by:  fctop 7610  cctop 7612

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-in 2048  df-ss 2050

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