Theorem rabid2 1746

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		pm4.71 633	. . . 4 |- ((x e. A -> ph) <-> (x e. A <-> (x e. A /\ ph)))
2	1	albii 975	. . 3 |- (A.x(x e. A -> ph) <-> A.x(x e. A <-> (x e. A /\ ph)))
3		abeq2 1544	. . 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 1625	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
6		df-rab 1628	. . 3 |- {x e. A | ph} = {x | (x e. A /\ ph)}
7	6	eqeq2i 1461	. 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 950   = wceq 1099   e. wcel 1105  {cab 1440  A.wral 1621  {crab 1624

This theorem is referenced by:  class2seteq 2703  zfrep6 3554  abrexex 3799  ioomax 6276

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-ral 1625  df-rab 1628

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