Theorem eqsstr3d 3306

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 2358	. 2 |- (ph -> A = B)
3		eqsstr3d.2	. 2 |- (ph -> B C_ C)
4	2, 3	eqsstrd 3305	1 |- (ph -> A C_ C)

Syntax hints:   -> wi 4   = wceq 1642   C_ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  ssxpb  5055  ffvresb  5431