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Theorem bnj1154 31713
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 31466 . 2  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e. 
_V )  ->  ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) ) )
2 elisset 2973 . . . . 5  |-  ( B  e.  _V  ->  E. b 
b  =  B )
32bnj708 31471 . . . 4  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e. 
_V )  ->  E. b 
b  =  B )
4 df-fr 4666 . . . . . . . 8  |-  ( R  Fr  A  <->  A. b
( ( b  C_  A  /\  b  =/=  (/) )  ->  E. x  e.  b  A. y  e.  b  -.  y R x ) )
54biimpi 194 . . . . . . 7  |-  ( R  Fr  A  ->  A. b
( ( b  C_  A  /\  b  =/=  (/) )  ->  E. x  e.  b  A. y  e.  b  -.  y R x ) )
6519.21bi 1803 . . . . . 6  |-  ( R  Fr  A  ->  (
( b  C_  A  /\  b  =/=  (/) )  ->  E. x  e.  b  A. y  e.  b  -.  y R x ) )
763impib 1178 . . . . 5  |-  ( ( R  Fr  A  /\  b  C_  A  /\  b  =/=  (/) )  ->  E. x  e.  b  A. y  e.  b  -.  y R x )
8 sseq1 3365 . . . . . . 7  |-  ( b  =  B  ->  (
b  C_  A  <->  B  C_  A
) )
9 neeq1 2606 . . . . . . 7  |-  ( b  =  B  ->  (
b  =/=  (/)  <->  B  =/=  (/) ) )
108, 93anbi23d 1285 . . . . . 6  |-  ( b  =  B  ->  (
( R  Fr  A  /\  b  C_  A  /\  b  =/=  (/) )  <->  ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) ) ) )
11 raleq 2907 . . . . . . 7  |-  ( b  =  B  ->  ( A. y  e.  b  -.  y R x  <->  A. y  e.  B  -.  y R x ) )
1211rexeqbi1dv 2916 . . . . . 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 320 . . . . 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 211 . . . 4  |-  ( b  =  B  ->  (
( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x ) )
153, 14bnj593 31460 . . 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 31488 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 958   A.wal 1360    = wceq 1362   E.wex 1589    e. wcel 1755    =/= wne 2596   A.wral 2705   E.wrex 2706   _Vcvv 2962    C_ wss 3316   (/)c0 3625   class class class wbr 4280    Fr wfr 4663    /\ w-bnj17 31397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-v 2964  df-in 3323  df-ss 3330  df-fr 4666  df-bnj17 31398
This theorem is referenced by:  bnj1190  31722
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