Theorem intmin4 4278

Description: Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)

Assertion

Ref	Expression
intmin4	|- ( A C_ |^| { x | ph } -> |^| { x | ( A C_ x /\ ph ) } = |^| { x | ph } )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem intmin4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssintab 4265	. . . 4 |- ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) )
2		simpr 467	. . . . . . . 8 |- ( ( A C_ x /\ ph ) -> ph )
3		ancr 556	. . . . . . . 8 |- ( ( ph -> A C_ x ) -> ( ph -> ( A C_ x /\ ph ) ) )
4	2, 3	impbid2 209	. . . . . . 7 |- ( ( ph -> A C_ x ) -> ( ( A C_ x /\ ph ) <-> ph ) )
5	4	imbi1d 323	. . . . . 6 |- ( ( ph -> A C_ x ) -> ( ( ( A C_ x /\ ph ) -> y e. x ) <-> ( ph -> y e. x ) ) )
6	5	alimi 1695	. . . . 5 |- ( A. x ( ph -> A C_ x ) -> A. x ( ( ( A C_ x /\ ph ) -> y e. x ) <-> ( ph -> y e. x ) ) )
7		albi 1701	. . . . 5 |- ( A. x ( ( ( A C_ x /\ ph ) -> y e. x ) <-> ( ph -> y e. x ) ) -> ( A. x ( ( A C_ x /\ ph ) -> y e. x ) <-> A. x ( ph -> y e. x ) ) )
8	6, 7	syl 17	. . . 4 |- ( A. x ( ph -> A C_ x ) -> ( A. x ( ( A C_ x /\ ph ) -> y e. x ) <-> A. x ( ph -> y e. x ) ) )
9	1, 8	sylbi 200	. . 3 |- ( A C_ |^| { x | ph } -> ( A. x ( ( A C_ x /\ ph ) -> y e. x ) <-> A. x ( ph -> y e. x ) ) )
10		vex 3060	. . . 4 |- y e. _V
11	10	elintab 4259	. . 3 |- ( y e. |^| { x | ( A C_ x /\ ph ) } <-> A. x ( ( A C_ x /\ ph ) -> y e. x ) )
12	10	elintab 4259	. . 3 |- ( y e. |^| { x | ph } <-> A. x ( ph -> y e. x ) )
13	9, 11, 12	3bitr4g 296	. 2 |- ( A C_ |^| { x | ph } -> ( y e. |^| { x | ( A C_ x /\ ph ) } <-> y e. |^| { x | ph } ) )
14	13	eqrdv 2460	1 |- ( A C_ |^| { x | ph } -> |^| { x | ( A C_ x /\ ph ) } = |^| { x | ph } )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1453   = wceq 1455   e. wcel 1898   {cab 2448   C_ wss 3416   |^|cint 4248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-v 3059  df-in 3423  df-ss 3430  df-int 4249

This theorem is referenced by: (None)