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Theorem raldifsnb 4092
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
StepHypRef Expression
1 elsn 3975 . . . . . 6  |-  ( x  e.  { Y }  <->  x  =  Y )
2 nnel 2741 . . . . . 6  |-  ( -.  x  e/  { Y } 
<->  x  e.  { Y } )
3 nne 2597 . . . . . 6  |-  ( -.  x  =/=  Y  <->  x  =  Y )
41, 2, 33bitr4ri 278 . . . . 5  |-  ( -.  x  =/=  Y  <->  -.  x  e/  { Y } )
54con4bii 295 . . . 4  |-  ( x  =/=  Y  <->  x  e/  { Y } )
65imbi1i 323 . . 3  |-  ( ( x  =/=  Y  ->  ph )  <->  ( x  e/  { Y }  ->  ph )
)
76ralbii 2827 . 2  |-  ( A. x  e.  A  (
x  =/=  Y  ->  ph )  <->  A. x  e.  A  ( x  e/  { Y }  ->  ph ) )
8 raldifb 3575 . 2  |-  ( A. x  e.  A  (
x  e/  { Y }  ->  ph )  <->  A. x  e.  ( A  \  { Y } ) ph )
97, 8bitri 249 1  |-  ( A. x  e.  A  (
x  =/=  Y  ->  ph )  <->  A. x  e.  ( A  \  { Y } ) ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1399    e. wcel 1836    =/= wne 2591    e/ wnel 2592   A.wral 2746    \ cdif 3403   {csn 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-v 3053  df-dif 3409  df-sn 3962
This theorem is referenced by:  dff14b  6099
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