Theorem sseqtr4d 3036

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtr4d.1	|- ( ph -> A C_ B )
sseqtr4d.2	|- ( ph -> C = B )

Assertion

Ref	Expression
sseqtr4d	|- ( ph -> A C_ C )

Proof of Theorem sseqtr4d

Step	Hyp	Ref	Expression
1		sseqtr4d.1	. 2 |- ( ph -> A C_ B )
2		sseqtr4d.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2086	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3035	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1284   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  syl5sseqr  3048  fnfvima  5414  tfrlemiubacc  5967  rdgivallem  5991