Theorem dfrab2 2270

Description: Alternate definition of restricted class abstraction.

Assertion

Ref	Expression
dfrab2	|- {x e. A | ph} = ({x | ph} i^i A)

Distinct variable group:   x,A

Proof of Theorem dfrab2

Step	Hyp	Ref	Expression
1		df-rab 1649	. 2 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2		inab 2264	. . 3 |- ({x | x e. A} i^i {x | ph}) = {x | (x e. A /\ ph)}
3		abid2 1577	. . . 4 |- {x | x e. A} = A
4	3	ineq1i 2209	. . 3 |- ({x | x e. A} i^i {x | ph}) = (A i^i {x | ph})
5	2, 4	eqtr3 1494	. 2 |- {x | (x e. A /\ ph)} = (A i^i {x | ph})
6		incom 2204	. 2 |- (A i^i {x | ph}) = ({x | ph} i^i A)
7	1, 5, 6	3eqtr 1496	1 |- {x e. A | ph} = ({x | ph} i^i A)

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  {crab 1645   i^i cin 2042

This theorem is referenced by:  dfbdop2 9726

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-rab 1649  df-v 1808  df-in 2047

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