Theorem 3sstr4d 2104

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

Hypotheses

Ref	Expression
3sstr4d.1	|- (ph -> A (_ B)
3sstr4d.2	|- (ph -> C = A)
3sstr4d.3	|- (ph -> D = B)

Assertion

Ref	Expression
3sstr4d	|- (ph -> C (_ D)

Proof of Theorem 3sstr4d

Step	Hyp	Ref	Expression
1		3sstr4d.1	. 2 |- (ph -> A (_ B)
2		3sstr4d.2	. . 3 |- (ph -> C = A)
3		3sstr4d.3	. . 3 |- (ph -> D = B)
4	2, 3	sseq12d 2090	. 2 |- (ph -> (C (_ D <-> A (_ B))
5	1, 4	mpbird 196	1 |- (ph -> C (_ D)

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

This theorem is referenced by:  omwordri 4203  oewordri 4219  mapss 4346  iooss1 6373  iooss2 6374  fzss1t 6503  fzss2t 6504  clsss 7687  ntrss 7688  tgioolem 7914  occont 9160  spanss 9318

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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