Theorem syl6ss 1546

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 1525	. 2 |- (A (_ B <-> A (_ C)
4	1, 3	sylib 173	1 |- (ph -> A (_ C)

Syntax hints:   -> wi 2   = wceq 1091   (_ wss 1487

This theorem is referenced by:  syl6ssr 1547  cflecard 3707  infxpidmlem11 4943  dfchsup2 5299  hsupval2t 5301  hsupvalt 5302  shsupclt 5307  shsupunss 5316  shslub 5359  orthin 5371  h1datom 5483

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492

metamath.org