Theorem ssabral 2119

Description: The relation for a subclass of a class abstraction is equivalent to restricted quantification.

Assertion

Ref	Expression
ssabral	|- (A (_ {x | ph} <-> A.x e. A ph)

Distinct variable group:   x,A

Proof of Theorem ssabral

Step	Hyp	Ref	Expression
1		ssab 2118	. 2 |- (A (_ {x | ph} <-> A.x(x e. A -> ph))
2		df-ral 1649	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
3	1, 2	bitr4 176	1 |- (A (_ {x | ph} <-> A.x e. A ph)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   e. wcel 958  {cab 1463  A.wral 1645   (_ wss 2047

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-ral 1649  df-in 2051  df-ss 2053

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