Theorem dfrab2 3903

Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem dfrab2

Step	Hyp	Ref	Expression
1		dfrab3 3902	. 2 |- { x e. A | ph } = ( A i^i { x | ph } )
2		incom 3805	. 2 |- ( A i^i { x | ph } ) = ( { x | ph } i^i A )
3	1, 2	eqtri 2644	1 |- { x e. A | ph } = ( { x | ph } i^i A )

Syntax hints:   = wceq 1483   {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:  dfpred3  5690  lubdm  16979  glbdm  16992  psrbagsn  19495  ismbl  23294  eulerpartgbij  30434  orvcval4  30522  fvline2  32253  abeqin  34017  nznngen  38515