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Metamath Proof Explorer

Theorem raldifsni

Description: Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015)

Ref Expression
Assertion raldifsni 
|- ( A. x e. ( A \ { B } ) -. ph <-> A. x e. A ( ph -> x = B ) )

Proof

Step Hyp Ref Expression
1 eldifsn 
 |-  ( x e. ( A \ { B } ) <-> ( x e. A /\ x =/= B ) )
2 1 imbi1i 
 |-  ( ( x e. ( A \ { B } ) -> -. ph ) <-> ( ( x e. A /\ x =/= B ) -> -. ph ) )
3 impexp 
 |-  ( ( ( x e. A /\ x =/= B ) -> -. ph ) <-> ( x e. A -> ( x =/= B -> -. ph ) ) )
4 df-ne 
 |-  ( x =/= B <-> -. x = B )
5 4 imbi1i 
 |-  ( ( x =/= B -> -. ph ) <-> ( -. x = B -> -. ph ) )
6 con34b 
 |-  ( ( ph -> x = B ) <-> ( -. x = B -> -. ph ) )
7 5 6 bitr4i 
 |-  ( ( x =/= B -> -. ph ) <-> ( ph -> x = B ) )
8 7 imbi2i 
 |-  ( ( x e. A -> ( x =/= B -> -. ph ) ) <-> ( x e. A -> ( ph -> x = B ) ) )
9 2 3 8 3bitri 
 |-  ( ( x e. ( A \ { B } ) -> -. ph ) <-> ( x e. A -> ( ph -> x = B ) ) )
10 9 ralbii2 
 |-  ( A. x e. ( A \ { B } ) -. ph <-> A. x e. A ( ph -> x = B ) )