Theorem rabbi 2789

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2850. (Contributed by NM, 25-Nov-2013.)

Assertion

Ref	Expression
rabbi	|- (A.x e. A (ps <-> ch) <-> {x e. A | ps} = {x e. A | ch})

Proof of Theorem rabbi

Step	Hyp	Ref	Expression
1		abbi 2463	. 2 |- (A.x((x e. A /\ ps) <-> (x e. A /\ ch)) <-> {x | (x e. A /\ ps)} = {x | (x e. A /\ ch)})
2		df-ral 2619	. . 3 |- (A.x e. A (ps <-> ch) <-> A.x(x e. A -> (ps <-> ch)))
3		pm5.32 617	. . . 4 |- ((x e. A -> (ps <-> ch)) <-> ((x e. A /\ ps) <-> (x e. A /\ ch)))
4	3	albii 1566	. . 3 |- (A.x(x e. A -> (ps <-> ch)) <-> A.x((x e. A /\ ps) <-> (x e. A /\ ch)))
5	2, 4	bitri 240	. 2 |- (A.x e. A (ps <-> ch) <-> A.x((x e. A /\ ps) <-> (x e. A /\ ch)))
6		df-rab 2623	. . 3 |- {x e. A | ps} = {x | (x e. A /\ ps)}
7		df-rab 2623	. . 3 |- {x e. A | ch} = {x | (x e. A /\ ch)}
8	6, 7	eqeq12i 2366	. 2 |- ({x e. A | ps} = {x e. A | ch} <-> {x | (x e. A /\ ps)} = {x | (x e. A /\ ch)})
9	1, 5, 8	3bitr4i 268	1 |- (A.x e. A (ps <-> ch) <-> {x e. A | ps} = {x e. A | ch})

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2614  {crab 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-ral 2619  df-rab 2623

This theorem is referenced by:  rabbidva  2850  fnpm  6008