Theorem 3sstr3g 3189

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Hypotheses

Ref	Expression
3sstr3g.1	|- ( ph -> A C_ B )
3sstr3g.2	|- A = C
3sstr3g.3	|- B = D

Assertion

Ref	Expression
3sstr3g	|- ( ph -> C C_ D )

Proof of Theorem 3sstr3g

Step	Hyp	Ref	Expression
1		3sstr3g.1	. 2 |- ( ph -> A C_ B )
2		3sstr3g.2	. . 3 |- A = C
3		3sstr3g.3	. . 3 |- B = D
4	2, 3	sseq12i 3175	. 2 |- ( A C_ B <-> C C_ D )
5	1, 4	sylib 121	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1348   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  hmeontr  13107