Theorem eqsstrd 2095

Description: Substitution of equality into a subclass relationship.

Hypotheses

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

Assertion

Ref	Expression
eqsstrd	|- (ph -> A (_ C)

Proof of Theorem eqsstrd

Step	Hyp	Ref	Expression
1		eqsstrd.2	. 2 |- (ph -> B (_ C)
2		eqsstrd.1	. . 3 |- (ph -> A = B)
3	2	sseq1d 2088	. 2 |- (ph -> (A (_ C <-> B (_ C))
4	1, 3	mpbird 196	1 |- (ph -> A (_ C)

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

This theorem is referenced by:  eqsstr3d 2096  snsspr 2470  fimacnv 3810  oawordeulem 4188  oewordri 4219  oaabslem 4251  mapsspw 4341  fodomr 4483  r1val1 4658  cardonle 4822  carduniima 4890  cfub 4908  cflecard 4912  uzssz 6430  infxpidmlem7 7558  infxpidmlem8 7559  subbas2OLD 7645  ntrss2 7690  lpsscls 7745  cnconst 7780  blssm 7850  rnblssm 7851  opnfss 7858  tgioolem 7914  chssoct 9419  specclt 9825  elnlfn2t 9853  mdsl0 10237  mdexch 10262  atcvat3 10323  dmdbr5at 10349  clsrebb 10493  subsp 10554

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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