Theorem intmin4 3898

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 3887	. . . 4 |- ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) )
2		simpr 110	. . . . . . . 8 |- ( ( A C_ x /\ ph ) -> ph )
3		ancr 321	. . . . . . . 8 |- ( ( ph -> A C_ x ) -> ( ph -> ( A C_ x /\ ph ) ) )
4	2, 3	impbid2 143	. . . . . . 7 |- ( ( ph -> A C_ x ) -> ( ( A C_ x /\ ph ) <-> ph ) )
5	4	imbi1d 231	. . . . . 6 |- ( ( ph -> A C_ x ) -> ( ( ( A C_ x /\ ph ) -> y e. x ) <-> ( ph -> y e. x ) ) )
6	5	alimi 1466	. . . . 5 |- ( A. x ( ph -> A C_ x ) -> A. x ( ( ( A C_ x /\ ph ) -> y e. x ) <-> ( ph -> y e. x ) ) )
7		albi 1479	. . . . 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 14	. . . 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 121	. . 3 |- ( A C_ |^| { x | ph } -> ( A. x ( ( A C_ x /\ ph ) -> y e. x ) <-> A. x ( ph -> y e. x ) ) )
10		vex 2763	. . . 4 |- y e. _V
11	10	elintab 3881	. . 3 |- ( y e. |^| { x | ( A C_ x /\ ph ) } <-> A. x ( ( A C_ x /\ ph ) -> y e. x ) )
12	10	elintab 3881	. . 3 |- ( y e. |^| { x | ph } <-> A. x ( ph -> y e. x ) )
13	9, 11, 12	3bitr4g 223	. 2 |- ( A C_ |^| { x | ph } -> ( y e. |^| { x | ( A C_ x /\ ph ) } <-> y e. |^| { x | ph } ) )
14	13	eqrdv 2191	1 |- ( A C_ |^| { x | ph } -> |^| { x | ( A C_ x /\ ph ) } = |^| { x | ph } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2164   {cab 2179   C_ wss 3153   |^|cint 3870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-in 3159  df-ss 3166  df-int 3871

This theorem is referenced by: (None)