Theorem rabid2 1309

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(s):   x,A

Proof of Theorem rabid2

Step	Hyp	Ref	Expression
1		pm4.71 481	. . . 4 |- ((x e. A -> ph) <-> (x e. A <-> (x e. A /\ ph)))
2	1	bial 695	. . 3 |- (A.x(x e. A -> ph) <-> A.x(x e. A <-> (x e. A /\ ph)))
3		cleqab 1174	. . 3 |- (A = {x | (x e. A /\ ph)} <-> A.x(x e. A <-> (x e. A /\ ph)))
4	2, 3	bitr4 154	. 2 |- (A.x(x e. A -> ph) <-> A = {x | (x e. A /\ ph)})
5		df-ral 1205	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
6		df-rab 1208	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
7	6	cleq2i 1111	. 2 |- (A = {x e. A | ph} <-> A = {x | (x e. A /\ ph)})
8	4, 5, 7	3bitr4r 159	1 |- (A = {x e. A | ph} <-> A.x e. A ph)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  {crab 1204

This theorem is referenced by:  zfrep6 2744  abrexex 2912

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-ral 1205  df-rab 1208

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