Theorem intmin4 3794

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 3783	. . . 4 |- ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) )
2		simpr 109	. . . . . . . 8 |- ( ( A C_ x /\ ph ) -> ph )
3		ancr 319	. . . . . . . 8 |- ( ( ph -> A C_ x ) -> ( ph -> ( A C_ x /\ ph ) ) )
4	2, 3	impbid2 142	. . . . . . 7 |- ( ( ph -> A C_ x ) -> ( ( A C_ x /\ ph ) <-> ph ) )
5	4	imbi1d 230	. . . . . 6 |- ( ( ph -> A C_ x ) -> ( ( ( A C_ x /\ ph ) -> y e. x ) <-> ( ph -> y e. x ) ) )
6	5	alimi 1431	. . . . 5 |- ( A. x ( ph -> A C_ x ) -> A. x ( ( ( A C_ x /\ ph ) -> y e. x ) <-> ( ph -> y e. x ) ) )
7		albi 1444	. . . . 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 120	. . 3 |- ( A C_ |^| { x | ph } -> ( A. x ( ( A C_ x /\ ph ) -> y e. x ) <-> A. x ( ph -> y e. x ) ) )
10		vex 2684	. . . 4 |- y e. _V
11	10	elintab 3777	. . 3 |- ( y e. |^| { x | ( A C_ x /\ ph ) } <-> A. x ( ( A C_ x /\ ph ) -> y e. x ) )
12	10	elintab 3777	. . 3 |- ( y e. |^| { x | ph } <-> A. x ( ph -> y e. x ) )
13	9, 11, 12	3bitr4g 222	. 2 |- ( A C_ |^| { x | ph } -> ( y e. |^| { x | ( A C_ x /\ ph ) } <-> y e. |^| { x | ph } ) )
14	13	eqrdv 2135	1 |- ( A C_ |^| { x | ph } -> |^| { x | ( A C_ x /\ ph ) } = |^| { x | ph } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   = wceq 1331   e. wcel 1480   {cab 2123   C_ wss 3066   |^|cint 3766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-int 3767

This theorem is referenced by: (None)