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Theorem bnj1154 28707
Description: Property of  Fr. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1154  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e. 
_V )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
Distinct variable groups:    x, A, y    x, B, y    x, R, y

Proof of Theorem bnj1154
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 bnj658 28458 . 2  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e. 
_V )  ->  ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) ) )
2 elisset 2910 . . . . 5  |-  ( B  e.  _V  ->  E. b 
b  =  B )
32bnj708 28463 . . . 4  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e. 
_V )  ->  E. b 
b  =  B )
4 df-fr 4483 . . . . . . . 8  |-  ( R  Fr  A  <->  A. b
( ( b  C_  A  /\  b  =/=  (/) )  ->  E. x  e.  b  A. y  e.  b  -.  y R x ) )
54biimpi 187 . . . . . . 7  |-  ( R  Fr  A  ->  A. b
( ( b  C_  A  /\  b  =/=  (/) )  ->  E. x  e.  b  A. y  e.  b  -.  y R x ) )
6519.21bi 1766 . . . . . 6  |-  ( R  Fr  A  ->  (
( b  C_  A  /\  b  =/=  (/) )  ->  E. x  e.  b  A. y  e.  b  -.  y R x ) )
763impib 1151 . . . . 5  |-  ( ( R  Fr  A  /\  b  C_  A  /\  b  =/=  (/) )  ->  E. x  e.  b  A. y  e.  b  -.  y R x )
8 sseq1 3313 . . . . . . 7  |-  ( b  =  B  ->  (
b  C_  A  <->  B  C_  A
) )
9 neeq1 2559 . . . . . . 7  |-  ( b  =  B  ->  (
b  =/=  (/)  <->  B  =/=  (/) ) )
108, 93anbi23d 1257 . . . . . 6  |-  ( b  =  B  ->  (
( R  Fr  A  /\  b  C_  A  /\  b  =/=  (/) )  <->  ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) ) ) )
11 raleq 2848 . . . . . . 7  |-  ( b  =  B  ->  ( A. y  e.  b  -.  y R x  <->  A. y  e.  B  -.  y R x ) )
1211rexeqbi1dv 2857 . . . . . 6  |-  ( b  =  B  ->  ( E. x  e.  b  A. y  e.  b  -.  y R x  <->  E. x  e.  B  A. y  e.  B  -.  y R x ) )
1310, 12imbi12d 312 . . . . 5  |-  ( b  =  B  ->  (
( ( R  Fr  A  /\  b  C_  A  /\  b  =/=  (/) )  ->  E. x  e.  b  A. y  e.  b  -.  y R x )  <-> 
( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) ) )
147, 13mpbii 203 . . . 4  |-  ( b  =  B  ->  (
( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) )
153, 14bnj593 28452 . . 3  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e. 
_V )  ->  E. b
( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) )
1615bnj937 28481 . 2  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e. 
_V )  ->  (
( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) )
171, 16mpd 15 1  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e. 
_V )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   E.wrex 2651   _Vcvv 2900    C_ wss 3264   (/)c0 3572   class class class wbr 4154    Fr wfr 4480    /\ w-bnj17 28389
This theorem is referenced by:  bnj1190  28716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-v 2902  df-in 3271  df-ss 3278  df-fr 4483  df-bnj17 28390
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