Theorem intmin4 3874

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 3863	. . . 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 1455	. . . . 5 |- ( A. x ( ph -> A C_ x ) -> A. x ( ( ( A C_ x /\ ph ) -> y e. x ) <-> ( ph -> y e. x ) ) )
7		albi 1468	. . . . 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 2742	. . . 4 |- y e. _V
11	10	elintab 3857	. . 3 |- ( y e. |^| { x | ( A C_ x /\ ph ) } <-> A. x ( ( A C_ x /\ ph ) -> y e. x ) )
12	10	elintab 3857	. . 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 2175	1 |- ( A C_ |^| { x | ph } -> |^| { x | ( A C_ x /\ ph ) } = |^| { x | ph } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2148   {cab 2163   C_ wss 3131   |^|cint 3846

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-in 3137  df-ss 3144  df-int 3847

This theorem is referenced by: (None)