Theorem syl5ss 1544

Description: A chained subclass and equality deduction.

Hypotheses

Ref	Expression
syl5ss.1	|- (ph -> A (_ B)
syl5ss.2	|- C = A

Assertion

Ref	Expression
syl5ss	|- (ph -> C (_ B)

Proof of Theorem syl5ss

Step	Hyp	Ref	Expression
1		syl5ss.1	. 2 |- (ph -> A (_ B)
2		syl5ss.2	. . 3 |- C = A
3	2	sseq1i 1524	. 2 |- (C (_ B <-> A (_ B)
4	1, 3	sylibr 175	1 |- (ph -> C (_ B)

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

This theorem is referenced by:  syl5ssr 1545  suceloni 2314  xpex 2488  cotr 2625  cnvsym 2626  fun 2763  fopab2 2891  oe0m1 3129  rankr1 3518  rankr1id 3539  oncard 3636  cflecard 3707  peano5nn 4424  uzwo3lem1 4614  uzwo3lem2 4615  sh0let 5365

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

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