Theorem ssabral 2122

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 2121	. 2 |- (A (_ {x | ph} <-> A.x(x e. A -> ph))
2		df-ral 1652	. 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 956   e. wcel 960  {cab 1466  A.wral 1648   (_ wss 2050

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-in 2054  df-ss 2056

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