Theorem 3sstr4g 2980

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr4g.1	|- (ph -> A C_ B)
3sstr4g.2	|- C = A
3sstr4g.3	|- D = B

Assertion

Ref	Expression
3sstr4g	|- (ph -> C C_ D)

Proof of Theorem 3sstr4g

Step	Hyp	Ref	Expression
1		3sstr4g.1	. 2 |- (ph -> A C_ B)
2		3sstr4g.2	. . 3 |- C = A
3		3sstr4g.3	. . 3 |- D = B
4	2, 3	sseq12i 2965	. 2 |- ( C C_ D <-> A C_ B)
5	1, 4	sylibr 137	1 |- (ph -> C C_ D)

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:  rabss2  3017  unss2  3108  sslin  3157  ssopab2  4003  xpss12  4388  coss1  4434  coss2  4435  cnvss  4451  rnss  4507  ssres  4580  ssres2  4581  imass1  4643  imass2  4644  imadif  4922  imain  4924  ssoprab2  5503  suppssfv  5650  suppssov1  5651  tposss  5802