Theorem 3sstr3 2099

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 2087	. 2 |- (A (_ B <-> C (_ D)
5	1, 4	mpbi 189	1 |- C (_ D

Syntax hints:   = wceq 956   (_ wss 2047

This theorem is referenced by:  fctopOLD 7650  cctop 7652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053

Copyright terms: Public domain