Theorem 3sstr3i 3643

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
3sstr3.1	|- A C_ B
3sstr3.2	|- A = C
3sstr3.3	|- B = D

Assertion

Ref	Expression
3sstr3i	|- C C_ D

Proof of Theorem 3sstr3i

Step	Hyp	Ref	Expression
1		3sstr3.1	. 2 |- A C_ B
2		3sstr3.2	. . 3 |- A = C
3		3sstr3.3	. . 3 |- B = D
4	2, 3	sseq12i 3631	. 2 |- ( A C_ B <-> C C_ D )
5	1, 4	mpbi 220	1 |- C C_ D

Syntax hints:   = wceq 1483   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588

This theorem is referenced by:  odf1o2  17988  leordtval2  21016  uniiccvol  23348  ballotlem2  30550  cotrcltrcl  38017