Theorem eqsstr3d 2974

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

Hypotheses

Ref	Expression
eqsstr3d.1	|- (ph -> B = A)
eqsstr3d.2	|- (ph -> B C_ C)

Assertion

Ref	Expression
eqsstr3d	|- (ph -> A C_ C)

Proof of Theorem eqsstr3d

Step	Hyp	Ref	Expression
1		eqsstr3d.1	. . 3 |- (ph -> B = A)
2	1	eqcomd 2042	. 2 |- (ph -> A = B)
3		eqsstr3d.2	. 2 |- (ph -> B C_ C)
4	2, 3	eqsstrd 2973	1 |- (ph -> A C_ C)

Syntax hints:   -> wi 4   = wceq 1242   C_ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925

This theorem is referenced by:  ssxpbm  4699  ssxp1  4700  ssxp2  4701  suppssof1  5670  tfrlemiubacc  5885  oaword1  5989  archnqq  6400