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Theorem onfrALTlem5VD 28715
Description: Virtual deduction proof of onfrALTlem5 28347. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem5 28347 is onfrALTlem5VD 28715 without virtual deductions and was automatically derived from onfrALTlem5VD 28715.
1::  |-  a  e.  _V
2:1:  |-  ( a  i^i  x )  e.  _V
3:2:  |-  ( [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) )
4:3:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  -.  ( a  i^i  x )  =  (/) )
5::  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x  )  =  (/) )
6:4,5:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =/=  (/) )
7:2:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  /  b ]. -.  b  =  (/) )
8::  |-  ( b  =/=  (/)  <->  -.  b  =  (/) )
9:8:  |-  A. b ( b  =/=  (/)  <->  -.  b  =  (/) )
10:2,9:  |-  ( [. ( a  i^i  x )  /  b ]. b  =/=  (/)  <->  [. ( a  i^i  x )  /  b ]. -.  b  =  (/) )
11:7,10:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  /  b ]. b  =/=  (/) )
12:6,11:  |-  ( [. ( a  i^i  x )  /  b ]. b  =/=  (/)  <->  (  a  i^i  x )  =/=  (/) )
13:2:  |-  ( [. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x  )  <->  ( a  i^i  x )  C_  ( a  i^i  x ) )
14:12,13:  |-  ( ( [. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x )  /\  [. ( a  i^i  x )  /  b ]. b  =/=  (/) )  <->  ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) ) )
15:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  <->  ( [. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x )  /\  [. ( a  i^i  x )  /  b ]. b  =/=  (/) ) )
16:15,14:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  <->  ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) ) )
17:2:  |-  [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  (  [_ ( a  i^i  x )  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )
18:2:  |-  [_ ( a  i^i  x )  /  b ]_ b  =  ( a  i^i  x )
19:2:  |-  [_ ( a  i^i  x )  /  b ]_ y  =  y
20:18,19:  |-  ( [_ ( a  i^i  x )  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )  =  ( ( a  i^i  x )  i^i  y )
21:17,20:  |-  [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  ( (  a  i^i  x )  i^i  y )
22:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/)  <->  [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  [_ ( a  i^i  x )  /  b ]_  (/) )
23:2:  |-  [_ ( a  i^i  x )  /  b ]_ (/)  =  (/)
24:21,23:  |-  ( [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  [_ ( a  i^i  x )  /  b ]_ (/)  <->  ( ( a  i^i  x )  i^i  y )  =  (/) )
25:22,24:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/)  <->  ( ( a  i^i  x )  i^i  y )  =  (/) )
26:2:  |-  ( [. ( a  i^i  x )  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x ) )
27:25,26:  |-  ( ( [. ( a  i^i  x )  /  b ]. y  e.  b  /\  [.  ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/) )  <->  ( y  e.  ( a  i^i  x )  /\  ( (  a  i^i  x )  i^i  y )  =  (/) ) )
28:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  ( [. ( a  i^i  x )  /  b ]. y  e.  b  /\  [. ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/) ) )
29:27,28:  |-  ( [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
30:29:  |-  A. y ( [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
31:30:  |-  ( E. y [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
32::  |-  ( E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/)  <->  E. y ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/)  ) )
33:31,32:  |-  ( E. y [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) )
34:2:  |-  ( E. y [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  (  b  i^i  y )  =  (/) ) )
35:33,34:  |-  ( [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y  )  =  (/) )
36::  |-  ( E. y  e.  b ( b  i^i  y )  =  (/)  <->  E. y  ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
37:36:  |-  A. b ( E. y  e.  b ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
38:2,37:  |-  ( [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/)  <->  [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
39:35,38:  |-  ( [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/)  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) )
40:16,39:  |-  ( ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/) )  <->  ( ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
41:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b ( b  i^i  y )  =  (/) )  <->  ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/) ) )
qed:40,41:  |-  ( [. ( a  i^i  x )  /  b ]. ( ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b ( b  i^i  y )  =  (/) )  <->  ( ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x  ) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem5VD  |-  ( [. ( a  i^i  x
)  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) 
<->  ( ( ( a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  (
a  i^i  x )
( ( a  i^i  x )  i^i  y
)  =  (/) ) )
Distinct variable groups:    a, b,
y    x, b, y

Proof of Theorem onfrALTlem5VD
StepHypRef Expression
1 vex 2927 . . . 4  |-  a  e. 
_V
21inex1 4312 . . 3  |-  ( a  i^i  x )  e. 
_V
3 sbcimg 3170 . . 3  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) 
<->  ( [. ( a  i^i  x )  / 
b ]. ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  [. ( a  i^i  x )  / 
b ]. E. y  e.  b  ( b  i^i  y )  =  (/) ) ) )
42, 3e0_ 28602 . 2  |-  ( [. ( a  i^i  x
)  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) 
<->  ( [. ( a  i^i  x )  / 
b ]. ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  [. ( a  i^i  x )  / 
b ]. E. y  e.  b  ( b  i^i  y )  =  (/) ) )
5 sbcang 3172 . . . . 5  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. ( b  C_  (
a  i^i  x )  /\  b  =/=  (/) )  <->  ( [. ( a  i^i  x
)  /  b ]. b  C_  ( a  i^i  x )  /\  [. (
a  i^i  x )  /  b ]. b  =/=  (/) ) ) )
62, 5e0_ 28602 . . . 4  |-  ( [. ( a  i^i  x
)  /  b ]. ( b  C_  (
a  i^i  x )  /\  b  =/=  (/) )  <->  ( [. ( a  i^i  x
)  /  b ]. b  C_  ( a  i^i  x )  /\  [. (
a  i^i  x )  /  b ]. b  =/=  (/) ) )
7 sseq1 3337 . . . . . . 7  |-  ( b  =  ( a  i^i  x )  ->  (
b  C_  ( a  i^i  x )  <->  ( a  i^i  x )  C_  (
a  i^i  x )
) )
87sbcieg 3161 . . . . . 6  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. b  C_  ( a  i^i  x )  <->  ( a  i^i  x )  C_  (
a  i^i  x )
) )
92, 8e0_ 28602 . . . . 5  |-  ( [. ( a  i^i  x
)  /  b ]. b  C_  ( a  i^i  x )  <->  ( a  i^i  x )  C_  (
a  i^i  x )
)
10 sbcng 3169 . . . . . . . . 9  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ].  -.  b  =  (/)  <->  -.  [. (
a  i^i  x )  /  b ]. b  =  (/) ) )
1110bicomd 193 . . . . . . . 8  |-  ( ( a  i^i  x )  e.  _V  ->  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ].  -.  b  =  (/) ) )
122, 11e0_ 28602 . . . . . . 7  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ].  -.  b  =  (/) )
13 df-ne 2577 . . . . . . . . 9  |-  ( b  =/=  (/)  <->  -.  b  =  (/) )
1413ax-gen 1552 . . . . . . . 8  |-  A. b
( b  =/=  (/)  <->  -.  b  =  (/) )
15 sbcbi 28343 . . . . . . . 8  |-  ( ( a  i^i  x )  e.  _V  ->  ( A. b ( b  =/=  (/) 
<->  -.  b  =  (/) )  ->  ( [. (
a  i^i  x )  /  b ]. b  =/=  (/)  <->  [. ( a  i^i  x )  /  b ].  -.  b  =  (/) ) ) )
162, 14, 15e00 28598 . . . . . . 7  |-  ( [. ( a  i^i  x
)  /  b ]. b  =/=  (/)  <->  [. ( a  i^i  x )  /  b ].  -.  b  =  (/) )
1712, 16bitr4i 244 . . . . . 6  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ]. b  =/=  (/) )
18 eqsbc3 3168 . . . . . . . . 9  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) ) )
192, 18e0_ 28602 . . . . . . . 8  |-  ( [. ( a  i^i  x
)  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) )
2019notbii 288 . . . . . . 7  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  -.  (
a  i^i  x )  =  (/) )
21 df-ne 2577 . . . . . . 7  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x )  =  (/) )
2220, 21bitr4i 244 . . . . . 6  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =/=  (/) )
2317, 22bitr3i 243 . . . . 5  |-  ( [. ( a  i^i  x
)  /  b ]. b  =/=  (/)  <->  ( a  i^i  x )  =/=  (/) )
249, 23anbi12i 679 . . . 4  |-  ( (
[. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x )  /\  [. ( a  i^i  x
)  /  b ]. b  =/=  (/) )  <->  ( (
a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) ) )
256, 24bitri 241 . . 3  |-  ( [. ( a  i^i  x
)  /  b ]. ( b  C_  (
a  i^i  x )  /\  b  =/=  (/) )  <->  ( (
a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) ) )
26 df-rex 2680 . . . . . 6  |-  ( E. y  e.  b  ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  (
b  i^i  y )  =  (/) ) )
2726ax-gen 1552 . . . . 5  |-  A. b
( E. y  e.  b  ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
28 sbcbi 28343 . . . . 5  |-  ( ( a  i^i  x )  e.  _V  ->  ( A. b ( E. y  e.  b  ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )  ->  ( [. ( a  i^i  x
)  /  b ]. E. y  e.  b 
( b  i^i  y
)  =  (/)  <->  [. ( a  i^i  x )  / 
b ]. E. y ( y  e.  b  /\  ( b  i^i  y
)  =  (/) ) ) ) )
292, 27, 28e00 28598 . . . 4  |-  ( [. ( a  i^i  x
)  /  b ]. E. y  e.  b 
( b  i^i  y
)  =  (/)  <->  [. ( a  i^i  x )  / 
b ]. E. y ( y  e.  b  /\  ( b  i^i  y
)  =  (/) ) )
30 sbcexg 3179 . . . . . . 7  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y [. (
a  i^i  x )  /  b ]. (
y  e.  b  /\  ( b  i^i  y
)  =  (/) ) ) )
3130bicomd 193 . . . . . 6  |-  ( ( a  i^i  x )  e.  _V  ->  ( E. y [. ( a  i^i  x )  / 
b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  (
b  i^i  y )  =  (/) ) ) )
322, 31e0_ 28602 . . . . 5  |-  ( E. y [. ( a  i^i  x )  / 
b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  (
b  i^i  y )  =  (/) ) )
33 sbcang 3172 . . . . . . . . . 10  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) 
<->  ( [. ( a  i^i  x )  / 
b ]. y  e.  b  /\  [. ( a  i^i  x )  / 
b ]. ( b  i^i  y )  =  (/) ) ) )
342, 33e0_ 28602 . . . . . . . . 9  |-  ( [. ( a  i^i  x
)  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) 
<->  ( [. ( a  i^i  x )  / 
b ]. y  e.  b  /\  [. ( a  i^i  x )  / 
b ]. ( b  i^i  y )  =  (/) ) )
35 sbcel2gv 3189 . . . . . . . . . . 11  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x
) ) )
362, 35e0_ 28602 . . . . . . . . . 10  |-  ( [. ( a  i^i  x
)  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x
) )
37 sbceqg 3235 . . . . . . . . . . . 12  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. ( b  i^i  y
)  =  (/)  <->  [_ ( a  i^i  x )  / 
b ]_ ( b  i^i  y )  =  [_ ( a  i^i  x
)  /  b ]_ (/) ) )
382, 37e0_ 28602 . . . . . . . . . . 11  |-  ( [. ( a  i^i  x
)  /  b ]. ( b  i^i  y
)  =  (/)  <->  [_ ( a  i^i  x )  / 
b ]_ ( b  i^i  y )  =  [_ ( a  i^i  x
)  /  b ]_ (/) )
39 csbing 3516 . . . . . . . . . . . . . 14  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ (
b  i^i  y )  =  ( [_ (
a  i^i  x )  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y ) )
402, 39e0_ 28602 . . . . . . . . . . . . 13  |-  [_ (
a  i^i  x )  /  b ]_ (
b  i^i  y )  =  ( [_ (
a  i^i  x )  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )
41 csbvarg 3246 . . . . . . . . . . . . . . 15  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ b  =  ( a  i^i  x ) )
422, 41e0_ 28602 . . . . . . . . . . . . . 14  |-  [_ (
a  i^i  x )  /  b ]_ b  =  ( a  i^i  x )
43 csbconstg 3233 . . . . . . . . . . . . . . 15  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ y  =  y )
442, 43e0_ 28602 . . . . . . . . . . . . . 14  |-  [_ (
a  i^i  x )  /  b ]_ y  =  y
4542, 44ineq12i 3508 . . . . . . . . . . . . 13  |-  ( [_ ( a  i^i  x
)  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )  =  ( ( a  i^i  x
)  i^i  y )
4640, 45eqtri 2432 . . . . . . . . . . . 12  |-  [_ (
a  i^i  x )  /  b ]_ (
b  i^i  y )  =  ( ( a  i^i  x )  i^i  y )
47 csbconstg 3233 . . . . . . . . . . . . 13  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ (/)  =  (/) )
482, 47e0_ 28602 . . . . . . . . . . . 12  |-  [_ (
a  i^i  x )  /  b ]_ (/)  =  (/)
4946, 48eqeq12i 2425 . . . . . . . . . . 11  |-  ( [_ ( a  i^i  x
)  /  b ]_ ( b  i^i  y
)  =  [_ (
a  i^i  x )  /  b ]_ (/)  <->  ( (
a  i^i  x )  i^i  y )  =  (/) )
5038, 49bitri 241 . . . . . . . . . 10  |-  ( [. ( a  i^i  x
)  /  b ]. ( b  i^i  y
)  =  (/)  <->  ( (
a  i^i  x )  i^i  y )  =  (/) )
5136, 50anbi12i 679 . . . . . . . . 9  |-  ( (
[. ( a  i^i  x )  /  b ]. y  e.  b  /\  [. ( a  i^i  x )  /  b ]. ( b  i^i  y
)  =  (/) )  <->  ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) ) )
5234, 51bitri 241 . . . . . . . 8  |-  ( [. ( a  i^i  x
)  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) 
<->  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
5352ax-gen 1552 . . . . . . 7  |-  A. y
( [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) 
<->  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
54 exbi 1588 . . . . . . 7  |-  ( A. y ( [. (
a  i^i  x )  /  b ]. (
y  e.  b  /\  ( b  i^i  y
)  =  (/) )  <->  ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) ) )  ->  ( E. y [. ( a  i^i  x )  / 
b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y ( y  e.  ( a  i^i  x )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) ) ) )
5553, 54e0_ 28602 . . . . . 6  |-  ( E. y [. ( a  i^i  x )  / 
b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y ( y  e.  ( a  i^i  x )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) ) )
56 df-rex 2680 . . . . . 6  |-  ( E. y  e.  ( a  i^i  x ) ( ( a  i^i  x
)  i^i  y )  =  (/)  <->  E. y ( y  e.  ( a  i^i  x )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) ) )
5755, 56bitr4i 244 . . . . 5  |-  ( E. y [. ( a  i^i  x )  / 
b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y
)  =  (/) )
5832, 57bitr3i 243 . . . 4  |-  ( [. ( a  i^i  x
)  /  b ]. E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y
)  =  (/) )
5929, 58bitri 241 . . 3  |-  ( [. ( a  i^i  x
)  /  b ]. E. y  e.  b 
( b  i^i  y
)  =  (/)  <->  E. y  e.  ( a  i^i  x
) ( ( a  i^i  x )  i^i  y )  =  (/) )
6025, 59imbi12i 317 . 2  |-  ( (
[. ( a  i^i  x )  /  b ]. ( b  C_  (
a  i^i  x )  /\  b  =/=  (/) )  ->  [. ( a  i^i  x
)  /  b ]. E. y  e.  b 
( b  i^i  y
)  =  (/) )  <->  ( (
( a  i^i  x
)  C_  ( a  i^i  x )  /\  (
a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x
) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
614, 60bitri 241 1  |-  ( [. ( a  i^i  x
)  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) 
<->  ( ( ( a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  (
a  i^i  x )
( ( a  i^i  x )  i^i  y
)  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675   _Vcvv 2924   [.wsbc 3129   [_csb 3219    i^i cin 3287    C_ wss 3288   (/)c0 3596
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-in 3295  df-ss 3302
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