Theorem rabid2 1770

Description: An "identity" law for restricted class abstraction.

Assertion

Ref	Expression
rabid2	|- (A = {x e. A | ph} <-> A.x e. A ph)

Distinct variable group:   x,A

Proof of Theorem rabid2

Step	Hyp	Ref	Expression
1		pm4.71 635	. . . 4 |- ((x e. A -> ph) <-> (x e. A <-> (x e. A /\ ph)))
2	1	albii 999	. . 3 |- (A.x(x e. A -> ph) <-> A.x(x e. A <-> (x e. A /\ ph)))
3		abeq2 1568	. . 3 |- (A = {x | (x e. A /\ ph)} <-> A.x(x e. A <-> (x e. A /\ ph)))
4	2, 3	bitr4 176	. 2 |- (A.x(x e. A -> ph) <-> A = {x | (x e. A /\ ph)})
5		df-ral 1649	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
6		df-rab 1652	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
7	6	eqeq2i 1485	. 2 |- (A = {x e. A | ph} <-> A = {x | (x e. A /\ ph)})
8	4, 5, 7	3bitr4r 184	1 |- (A = {x e. A | ph} <-> A.x e. A ph)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  {crab 1648

This theorem is referenced by:  class2seteq 2735  zfrep6 3614  abrexex 3860  ioomax 6393

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

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