Theorem rabss 2124

Description: Restricted class abstraction in a subclass relationship.

Assertion

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

Distinct variable group:   x,B

Proof of Theorem rabss

Step	Hyp	Ref	Expression
1		df-rab 1652	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2	1	sseq1i 2085	. 2 |- ({x e. A | ph} (_ B <-> {x | (x e. A /\ ph)} (_ B)
3		abss 2117	. 2 |- ({x | (x e. A /\ ph)} (_ B <-> A.x((x e. A /\ ph) -> x e. B))
4		impexp 347	. . . 4 |- (((x e. A /\ ph) -> x e. B) <-> (x e. A -> (ph -> x e. B)))
5	4	albii 999	. . 3 |- (A.x((x e. A /\ ph) -> x e. B) <-> A.x(x e. A -> (ph -> x e. B)))
6		df-ral 1649	. . 3 |- (A.x e. A (ph -> x e. B) <-> A.x(x e. A -> (ph -> x e. B)))
7	5, 6	bitr4 176	. 2 |- (A.x((x e. A /\ ph) -> x e. B) <-> A.x e. A (ph -> x e. B))
8	2, 3, 7	3bitr 177	1 |- ({x e. A | ph} (_ B <-> A.x e. A (ph -> x e. B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   e. wcel 958  {cab 1463  A.wral 1645  {crab 1648   (_ 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-rab 1652  df-in 2051  df-ss 2053

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