Theorem dfrab3 3457

Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
dfrab3	|- { x e. A | ph } = ( A i^i { x | ph } )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem dfrab3

Step	Hyp	Ref	Expression
1		df-rab 2495	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		inab 3449	. 2 |- ( { x | x e. A } i^i { x | ph } ) = { x | ( x e. A /\ ph ) }
3		abid2 2328	. . 3 |- { x | x e. A } = A
4	3	ineq1i 3378	. 2 |- ( { x | x e. A } i^i { x | ph } ) = ( A i^i { x | ph } )
5	1, 2, 4	3eqtr2i 2234	1 |- { x e. A | ph } = ( A i^i { x | ph } )

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2178   {cab 2193   {crab 2490   i^i cin 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778  df-in 3180

This theorem is referenced by:  notrab  3458  dfrab3ss  3459  dfif3  3593  dfse2  5074