Theorem 3sstr4g 3226

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 3211	. 2 |- ( C C_ D <-> A C_ B )
5	1, 4	sylibr 134	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  rabss2  3266  unss2  3334  sslin  3389  ssopab2  4310  xpss12  4770  coss1  4821  coss2  4822  cnvss  4839  rnss  4896  ssres  4972  ssres2  4973  imass1  5044  imass2  5045  imadif  5338  imain  5340  ssoprab2  5978  suppssfv  6131  suppssov1  6132  tposss  6304  ss2ixp  6770  isumsplit  11656  isumrpcl  11659