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Theorem onfrALTlem5VD 37282
Description: Virtual deduction proof of onfrALTlem5 36908. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem5 36908 is onfrALTlem5VD 37282 without virtual deductions and was automatically derived from onfrALTlem5VD 37282.
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 3048 . . . 4  |-  a  e. 
_V
21inex1 4544 . . 3  |-  ( a  i^i  x )  e. 
_V
3 sbcimg 3309 . . 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, 3e0a 37159 . 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 sbcangOLD 36890 . . . . 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, 5e0a 37159 . . . 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 3453 . . . . . . 7  |-  ( b  =  ( a  i^i  x )  ->  (
b  C_  ( a  i^i  x )  <->  ( a  i^i  x )  C_  (
a  i^i  x )
) )
87sbcieg 3300 . . . . . 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, 8e0a 37159 . . . . 5  |-  ( [. ( a  i^i  x
)  /  b ]. b  C_  ( a  i^i  x )  <->  ( a  i^i  x )  C_  (
a  i^i  x )
)
10 sbcng 3308 . . . . . . . . 9  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ].  -.  b  =  (/)  <->  -.  [. (
a  i^i  x )  /  b ]. b  =  (/) ) )
1110bicomd 205 . . . . . . . 8  |-  ( ( a  i^i  x )  e.  _V  ->  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ].  -.  b  =  (/) ) )
122, 11e0a 37159 . . . . . . 7  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ].  -.  b  =  (/) )
13 df-ne 2624 . . . . . . . . 9  |-  ( b  =/=  (/)  <->  -.  b  =  (/) )
1413ax-gen 1669 . . . . . . . 8  |-  A. b
( b  =/=  (/)  <->  -.  b  =  (/) )
15 sbcbi 36900 . . . . . . . 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 37155 . . . . . . 7  |-  ( [. ( a  i^i  x
)  /  b ]. b  =/=  (/)  <->  [. ( a  i^i  x )  /  b ].  -.  b  =  (/) )
1712, 16bitr4i 256 . . . . . 6  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ]. b  =/=  (/) )
18 eqsbc3 3307 . . . . . . . . 9  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) ) )
192, 18e0a 37159 . . . . . . . 8  |-  ( [. ( a  i^i  x
)  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) )
2019notbii 298 . . . . . . 7  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  -.  (
a  i^i  x )  =  (/) )
21 df-ne 2624 . . . . . . 7  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x )  =  (/) )
2220, 21bitr4i 256 . . . . . 6  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =/=  (/) )
2317, 22bitr3i 255 . . . . 5  |-  ( [. ( a  i^i  x
)  /  b ]. b  =/=  (/)  <->  ( a  i^i  x )  =/=  (/) )
249, 23anbi12i 703 . . . 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 253 . . 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 2743 . . . . . 6  |-  ( E. y  e.  b  ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  (
b  i^i  y )  =  (/) ) )
2726ax-gen 1669 . . . . 5  |-  A. b
( E. y  e.  b  ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
28 sbcbi 36900 . . . . 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 37155 . . . 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 sbcexgOLD 36904 . . . . . . 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 205 . . . . . 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, 31e0a 37159 . . . . 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 sbcangOLD 36890 . . . . . . . . . 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, 33e0a 37159 . . . . . . . . 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 3327 . . . . . . . . . . 11  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x
) ) )
362, 35e0a 37159 . . . . . . . . . 10  |-  ( [. ( a  i^i  x
)  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x
) )
37 sbceqg 3773 . . . . . . . . . . . 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, 37e0a 37159 . . . . . . . . . . 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 csbingOLD 37215 . . . . . . . . . . . . . 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, 39e0a 37159 . . . . . . . . . . . . 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 3792 . . . . . . . . . . . . . . 15  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ b  =  ( a  i^i  x ) )
422, 41e0a 37159 . . . . . . . . . . . . . 14  |-  [_ (
a  i^i  x )  /  b ]_ b  =  ( a  i^i  x )
43 csbconstg 3376 . . . . . . . . . . . . . . 15  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ y  =  y )
442, 43e0a 37159 . . . . . . . . . . . . . 14  |-  [_ (
a  i^i  x )  /  b ]_ y  =  y
4542, 44ineq12i 3632 . . . . . . . . . . . . 13  |-  ( [_ ( a  i^i  x
)  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )  =  ( ( a  i^i  x
)  i^i  y )
4640, 45eqtri 2473 . . . . . . . . . . . 12  |-  [_ (
a  i^i  x )  /  b ]_ (
b  i^i  y )  =  ( ( a  i^i  x )  i^i  y )
47 csbconstg 3376 . . . . . . . . . . . . 13  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ (/)  =  (/) )
482, 47e0a 37159 . . . . . . . . . . . 12  |-  [_ (
a  i^i  x )  /  b ]_ (/)  =  (/)
4946, 48eqeq12i 2465 . . . . . . . . . . 11  |-  ( [_ ( a  i^i  x
)  /  b ]_ ( b  i^i  y
)  =  [_ (
a  i^i  x )  /  b ]_ (/)  <->  ( (
a  i^i  x )  i^i  y )  =  (/) )
5038, 49bitri 253 . . . . . . . . . 10  |-  ( [. ( a  i^i  x
)  /  b ]. ( b  i^i  y
)  =  (/)  <->  ( (
a  i^i  x )  i^i  y )  =  (/) )
5136, 50anbi12i 703 . . . . . . . . 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 253 . . . . . . . 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 1669 . . . . . . 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 1716 . . . . . . 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, 54e0a 37159 . . . . . 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 2743 . . . . . 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 256 . . . . 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 255 . . . 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 253 . . 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 328 . 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 253 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   E.wrex 2738   _Vcvv 3045   [.wsbc 3267   [_csb 3363    i^i cin 3403    C_ wss 3404   (/)c0 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-in 3411  df-ss 3418
This theorem is referenced by: (None)
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