Theorem intexrab 4823

Description: The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)

Assertion

Ref	Expression
intexrab	|- ( E. x e. A ph <-> |^| { x e. A | ph } e. _V )

Proof of Theorem intexrab

Step	Hyp	Ref	Expression
1		intexab 4822	. 2 |- ( E. x ( x e. A /\ ph ) <-> |^| { x | ( x e. A /\ ph ) } e. _V )
2		df-rex 2918	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3		df-rab 2921	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	inteqi 4479	. . 3 |- |^| { x e. A | ph } = |^| { x | ( x e. A /\ ph ) }
5	4	eleq1i 2692	. 2 |- ( |^| { x e. A | ph } e. _V <-> |^| { x | ( x e. A /\ ph ) } e. _V )
6	1, 2, 5	3bitr4i 292	1 |- ( E. x e. A ph <-> |^| { x e. A | ph } e. _V )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   _Vcvv 3200   |^|cint 4475

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-int 4476

This theorem is referenced by:  onintrab2  7002  rankf  8657  rankvalb  8660  cardf2  8769  tskmval  9661  lspval  18975  aspval  19328  clsval  20841  spanval  28192  rgspnval  37738