Theorem dfrab3 3902

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 2921	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		inab 3895	. 2 |- ( { x | x e. A } i^i { x | ph } ) = { x | ( x e. A /\ ph ) }
3		abid2 2745	. . 3 |- { x | x e. A } = A
4	3	ineq1i 3810	. 2 |- ( { x | x e. A } i^i { x | ph } ) = ( A i^i { x | ph } )
5	1, 2, 4	3eqtr2i 2650	1 |- { x e. A | ph } = ( A i^i { x | ph } )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-in 3581

This theorem is referenced by:  dfrab2  3903  notrab  3904  dfrab3ss  3905  dfif3  4100  dffr3  5498  dfse2  5499  tz6.26  5711  rabfi  8185  dfsup2  8350  ressmplbas2  19455  clsocv  23049  hasheuni  30147  bj-inrab3  32925  hashnzfz  38519