Theorem syl6ss 2107

Description: A chained subclass and equality deduction.

Hypotheses

Ref	Expression
syl6ss.1	|- (ph -> A (_ B)
syl6ss.2	|- B = C

Assertion

Ref	Expression
syl6ss	|- (ph -> A (_ C)

Proof of Theorem syl6ss

Step	Hyp	Ref	Expression
1		syl6ss.1	. 2 |- (ph -> A (_ B)
2		syl6ss.2	. . 3 |- B = C
3	2	sseq2i 2086	. 2 |- (A (_ B <-> A (_ C)
4	1, 3	sylib 198	1 |- (ph -> A (_ C)

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

This theorem is referenced by:  syl6ssr 2108  sspr 2475  sspwuni 2758  cflecard 4912  infxpidmlem11 7562  distop 7649  elcls 7704  uniopn 7861  opnuni 7868  tgioo 7915  lmsslem 7952  dfchsup2 9298  hsupval2t 9300  hsupvalt 9301  shsupclt 9306  shsupunss 9315  shslub 9358  orthin 9370  h1datom 9504  mdslj2 10247  mdslmd1lem1 10252  fgsb 10570  fgsbOLD 10571

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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