Theorem raldifsni 4324

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

Assertion

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

Proof of Theorem raldifsni

Step	Hyp	Ref	Expression
1		eldifsn 4317	. . . 4 |- ( x e. ( A \ { B } ) <-> ( x e. A /\ x =/= B ) )
2	1	imbi1i 339	. . 3 |- ( ( x e. ( A \ { B } ) -> -. ph ) <-> ( ( x e. A /\ x =/= B ) -> -. ph ) )
3		impexp 462	. . 3 |- ( ( ( x e. A /\ x =/= B ) -> -. ph ) <-> ( x e. A -> ( x =/= B -> -. ph ) ) )
4		df-ne 2795	. . . . . 6 |- ( x =/= B <-> -. x = B )
5	4	imbi1i 339	. . . . 5 |- ( ( x =/= B -> -. ph ) <-> ( -. x = B -> -. ph ) )
6		con34b 306	. . . . 5 |- ( ( ph -> x = B ) <-> ( -. x = B -> -. ph ) )
7	5, 6	bitr4i 267	. . . 4 |- ( ( x =/= B -> -. ph ) <-> ( ph -> x = B ) )
8	7	imbi2i 326	. . 3 |- ( ( x e. A -> ( x =/= B -> -. ph ) ) <-> ( x e. A -> ( ph -> x = B ) ) )
9	2, 3, 8	3bitri 286	. 2 |- ( ( x e. ( A \ { B } ) -> -. ph ) <-> ( x e. A -> ( ph -> x = B ) ) )
10	9	ralbii2 2978	1 |- ( A. x e. ( A \ { B } ) -. ph <-> A. x e. A ( ph -> x = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-sn 4178

This theorem is referenced by:  islindf4  20177  snlindsntor  42260