Theorem inintabd 37885

Description: Value of the intersection of class with the intersection of a non-empty class. (Contributed by RP, 13-Aug-2020.)

Hypothesis

Ref	Expression
inintabd.x	|- ( ph -> E. x ps )

Assertion

Ref	Expression
inintabd	|- ( ph -> ( A i^i |^| { x | ps } ) = |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ps ) } )

Distinct variable groups:   ps, w   x, w, A

Allowed substitution hints:   ph( x, w)   ps( x)

Proof of Theorem inintabd

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inintabd.x	. . . . . 6 |- ( ph -> E. x ps )
2		pm5.5 351	. . . . . 6 |- ( E. x ps -> ( ( E. x ps -> u e. A ) <-> u e. A ) )
3	1, 2	syl 17	. . . . 5 |- ( ph -> ( ( E. x ps -> u e. A ) <-> u e. A ) )
4	3	bicomd 213	. . . 4 |- ( ph -> ( u e. A <-> ( E. x ps -> u e. A ) ) )
5	4	anbi1d 741	. . 3 |- ( ph -> ( ( u e. A /\ A. x ( ps -> u e. x ) ) <-> ( ( E. x ps -> u e. A ) /\ A. x ( ps -> u e. x ) ) ) )
6		elinintab 37881	. . 3 |- ( u e. ( A i^i |^| { x | ps } ) <-> ( u e. A /\ A. x ( ps -> u e. x ) ) )
7		vex 3203	. . . 4 |- u e. _V
8		elinintrab 37883	. . . 4 |- ( u e. _V -> ( u e. |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ps ) } <-> ( ( E. x ps -> u e. A ) /\ A. x ( ps -> u e. x ) ) ) )
9	7, 8	ax-mp 5	. . 3 |- ( u e. |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ps ) } <-> ( ( E. x ps -> u e. A ) /\ A. x ( ps -> u e. x ) ) )
10	5, 6, 9	3bitr4g 303	. 2 |- ( ph -> ( u e. ( A i^i |^| { x | ps } ) <-> u e. |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ps ) } ) )
11	10	eqrdv 2620	1 |- ( ph -> ( A i^i |^| { x | ps } ) = |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ps ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   i^i cin 3573   ~Pcpw 4158   |^|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-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-3an 1039  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-ral 2917  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-int 4476

This theorem is referenced by:  xpinintabd  37886