Theorem 3sstr4g 2102

Description: Substitution of equality into both sides of a subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

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

Assertion

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

Proof of Theorem 3sstr4g

Step	Hyp	Ref	Expression
1		3sstr4g.1	. 2 |- (ph -> A (_ B)
2		3sstr4g.2	. . 3 |- C = A
3		3sstr4g.3	. . 3 |- D = B
4	2, 3	sseq12i 2087	. 2 |- (C (_ D <-> A (_ B)
5	1, 4	sylibr 200	1 |- (ph -> C (_ D)

Syntax hints:   -> wi 3   = wceq 956   (_ wss 2047

This theorem is referenced by:  unss2 2201  sslin 2235  uniss 2521  iunss1 2574  ssopab2 2822  cnvss 3291  rnss 3342  ssres 3385  ssres2 3386  imass1 3432  imass2 3433  metss 7824  subgrnss 8119  sspba 8386  shlej2 9349  chpssat 10290

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053

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