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Theorem bnj1154 29305
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 29056 . 2  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e. 
_V )  ->  ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) ) )
2 elisset 2958 . . . . 5  |-  ( B  e.  _V  ->  E. b 
b  =  B )
32bnj708 29061 . . . 4  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e. 
_V )  ->  E. b 
b  =  B )
4 df-fr 4533 . . . . . . . 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 1774 . . . . . 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 3361 . . . . . . 7  |-  ( b  =  B  ->  (
b  C_  A  <->  B  C_  A
) )
9 neeq1 2606 . . . . . . 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 2896 . . . . . . 7  |-  ( b  =  B  ->  ( A. y  e.  b  -.  y R x  <->  A. y  e.  B  -.  y R x ) )
1211rexeqbi1dv 2905 . . . . . 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 29050 . . 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 29079 . 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 1549   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   (/)c0 3620   class class class wbr 4204    Fr wfr 4530    /\ w-bnj17 28987
This theorem is referenced by:  bnj1190  29314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-fr 4533  df-bnj17 28988
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