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Theorem onfrALTlem5VD 28661
Description: Virtual deduction proof of onfrALTlem5 28307. 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 28307 is onfrALTlem5VD 28661 without virtual deductions and was automatically derived from onfrALTlem5VD 28661.
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 2791 . . . 4  |-  a  e. 
_V
21inex1 4155 . . 3  |-  ( a  i^i  x )  e. 
_V
3 sbcimg 3032 . . 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_ 28547 . 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 3034 . . . . 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_ 28547 . . . 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 3199 . . . . . . 7  |-  ( b  =  ( a  i^i  x )  ->  (
b  C_  ( a  i^i  x )  <->  ( a  i^i  x )  C_  (
a  i^i  x )
) )
87sbcieg 3023 . . . . . 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_ 28547 . . . . 5  |-  ( [. ( a  i^i  x
)  /  b ]. b  C_  ( a  i^i  x )  <->  ( a  i^i  x )  C_  (
a  i^i  x )
)
10 sbcng 3031 . . . . . . . . 9  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ].  -.  b  =  (/)  <->  -.  [. (
a  i^i  x )  /  b ]. b  =  (/) ) )
1110bicomd 192 . . . . . . . 8  |-  ( ( a  i^i  x )  e.  _V  ->  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ].  -.  b  =  (/) ) )
122, 11e0_ 28547 . . . . . . 7  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ].  -.  b  =  (/) )
13 df-ne 2448 . . . . . . . . 9  |-  ( b  =/=  (/)  <->  -.  b  =  (/) )
1413ax-gen 1533 . . . . . . . 8  |-  A. b
( b  =/=  (/)  <->  -.  b  =  (/) )
15 sbcbi 28303 . . . . . . . 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 28543 . . . . . . 7  |-  ( [. ( a  i^i  x
)  /  b ]. b  =/=  (/)  <->  [. ( a  i^i  x )  /  b ].  -.  b  =  (/) )
1712, 16bitr4i 243 . . . . . 6  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ]. b  =/=  (/) )
18 eqsbc3 3030 . . . . . . . . 9  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) ) )
192, 18e0_ 28547 . . . . . . . 8  |-  ( [. ( a  i^i  x
)  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) )
2019notbii 287 . . . . . . 7  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  -.  (
a  i^i  x )  =  (/) )
21 df-ne 2448 . . . . . . 7  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x )  =  (/) )
2220, 21bitr4i 243 . . . . . 6  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =/=  (/) )
2317, 22bitr3i 242 . . . . 5  |-  ( [. ( a  i^i  x
)  /  b ]. b  =/=  (/)  <->  ( a  i^i  x )  =/=  (/) )
249, 23anbi12i 678 . . . 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 240 . . 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 2549 . . . . . 6  |-  ( E. y  e.  b  ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  (
b  i^i  y )  =  (/) ) )
2726ax-gen 1533 . . . . 5  |-  A. b
( E. y  e.  b  ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
28 sbcbi 28303 . . . . 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 28543 . . . 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 3041 . . . . . . 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 192 . . . . . 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_ 28547 . . . . 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 3034 . . . . . . . . . 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_ 28547 . . . . . . . . 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 3051 . . . . . . . . . . 11  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x
) ) )
362, 35e0_ 28547 . . . . . . . . . 10  |-  ( [. ( a  i^i  x
)  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x
) )
37 sbceqg 3097 . . . . . . . . . . . 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_ 28547 . . . . . . . . . . 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 3376 . . . . . . . . . . . . . 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_ 28547 . . . . . . . . . . . . 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 3108 . . . . . . . . . . . . . . 15  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ b  =  ( a  i^i  x ) )
422, 41e0_ 28547 . . . . . . . . . . . . . 14  |-  [_ (
a  i^i  x )  /  b ]_ b  =  ( a  i^i  x )
43 csbconstg 3095 . . . . . . . . . . . . . . 15  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ y  =  y )
442, 43e0_ 28547 . . . . . . . . . . . . . 14  |-  [_ (
a  i^i  x )  /  b ]_ y  =  y
4542, 44ineq12i 3368 . . . . . . . . . . . . 13  |-  ( [_ ( a  i^i  x
)  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )  =  ( ( a  i^i  x
)  i^i  y )
4640, 45eqtri 2303 . . . . . . . . . . . 12  |-  [_ (
a  i^i  x )  /  b ]_ (
b  i^i  y )  =  ( ( a  i^i  x )  i^i  y )
47 csbconstg 3095 . . . . . . . . . . . . 13  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ (/)  =  (/) )
482, 47e0_ 28547 . . . . . . . . . . . 12  |-  [_ (
a  i^i  x )  /  b ]_ (/)  =  (/)
4946, 48eqeq12i 2296 . . . . . . . . . . 11  |-  ( [_ ( a  i^i  x
)  /  b ]_ ( b  i^i  y
)  =  [_ (
a  i^i  x )  /  b ]_ (/)  <->  ( (
a  i^i  x )  i^i  y )  =  (/) )
5038, 49bitri 240 . . . . . . . . . 10  |-  ( [. ( a  i^i  x
)  /  b ]. ( b  i^i  y
)  =  (/)  <->  ( (
a  i^i  x )  i^i  y )  =  (/) )
5136, 50anbi12i 678 . . . . . . . . 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 240 . . . . . . . 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 1533 . . . . . . 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 1568 . . . . . . 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_ 28547 . . . . . 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 2549 . . . . . 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 243 . . . . 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 242 . . . 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 240 . . 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 316 . 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 240 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 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788   [.wsbc 2991   [_csb 3081    i^i cin 3151    C_ wss 3152   (/)c0 3455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-in 3159  df-ss 3166
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