Theorem raldifsnb 4325

Description: Restricted universal quantification on a class difference with a singleton in terms of an implication. (Contributed by Alexander van der Vekens, 26-Jan-2018.)

Assertion

Ref	Expression
raldifsnb	|- ( A. x e. A ( x =/= Y -> ph ) <-> A. x e. ( A \ { Y } ) ph )

Distinct variable group:   x, Y

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem raldifsnb

Step	Hyp	Ref	Expression
1		velsn 4193	. . . . . 6 |- ( x e. { Y } <-> x = Y )
2		nnel 2906	. . . . . 6 |- ( -. x e/ { Y } <-> x e. { Y } )
3		nne 2798	. . . . . 6 |- ( -. x =/= Y <-> x = Y )
4	1, 2, 3	3bitr4ri 293	. . . . 5 |- ( -. x =/= Y <-> -. x e/ { Y } )
5	4	con4bii 311	. . . 4 |- ( x =/= Y <-> x e/ { Y } )
6	5	imbi1i 339	. . 3 |- ( ( x =/= Y -> ph ) <-> ( x e/ { Y } -> ph ) )
7	6	ralbii 2980	. 2 |- ( A. x e. A ( x =/= Y -> ph ) <-> A. x e. A ( x e/ { Y } -> ph ) )
8		raldifb 3750	. 2 |- ( A. x e. A ( x e/ { Y } -> ph ) <-> A. x e. ( A \ { Y } ) ph )
9	7, 8	bitri 264	1 |- ( A. x e. A ( x =/= Y -> ph ) <-> A. x e. ( A \ { Y } ) ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   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-nel 2898  df-ral 2917  df-v 3202  df-dif 3577  df-sn 4178

This theorem is referenced by:  dff14b  6528