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Metamath Proof Explorer

Theorem intmin4

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

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

Proof

Step Hyp Ref Expression
1 ssintab 
 |-  ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) )
2 simpr 
 |-  ( ( A C_ x /\ ph ) -> ph )
3 ancr 
 |-  ( ( ph -> A C_ x ) -> ( ph -> ( A C_ x /\ ph ) ) )
4 2 3 impbid2 
 |-  ( ( ph -> A C_ x ) -> ( ( A C_ x /\ ph ) <-> ph ) )
5 4 imbi1d 
 |-  ( ( ph -> A C_ x ) -> ( ( ( A C_ x /\ ph ) -> y e. x ) <-> ( ph -> y e. x ) ) )
6 5 alimi 
 |-  ( A. x ( ph -> A C_ x ) -> A. x ( ( ( A C_ x /\ ph ) -> y e. x ) <-> ( ph -> y e. x ) ) )
7 albi 
 |-  ( 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 
 |-  ( 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 
 |-  ( A C_ |^| { x | ph } -> ( A. x ( ( A C_ x /\ ph ) -> y e. x ) <-> A. x ( ph -> y e. x ) ) )
10 vex 
 |-  y e. _V
11 10 elintab 
 |-  ( y e. |^| { x | ( A C_ x /\ ph ) } <-> A. x ( ( A C_ x /\ ph ) -> y e. x ) )
12 10 elintab 
 |-  ( y e. |^| { x | ph } <-> A. x ( ph -> y e. x ) )
13 9 11 12 3bitr4g 
 |-  ( A C_ |^| { x | ph } -> ( y e. |^| { x | ( A C_ x /\ ph ) } <-> y e. |^| { x | ph } ) )
14 13 eqrdv 
 |-  ( A C_ |^| { x | ph } -> |^| { x | ( A C_ x /\ ph ) } = |^| { x | ph } )