Step |
Hyp |
Ref |
Expression |
1 |
|
nfsbc1v |
|- F/ y [. ( 1st ` ( 1st ` x ) ) / y ]. [. ( 2nd ` ( 1st ` x ) ) / z ]. [. ( 2nd ` x ) / w ]. ph |
2 |
|
nfcv |
|- F/_ z ( 1st ` ( 1st ` x ) ) |
3 |
|
nfsbc1v |
|- F/ z [. ( 2nd ` ( 1st ` x ) ) / z ]. [. ( 2nd ` x ) / w ]. ph |
4 |
2 3
|
nfsbcw |
|- F/ z [. ( 1st ` ( 1st ` x ) ) / y ]. [. ( 2nd ` ( 1st ` x ) ) / z ]. [. ( 2nd ` x ) / w ]. ph |
5 |
|
nfcv |
|- F/_ w ( 1st ` ( 1st ` x ) ) |
6 |
|
nfcv |
|- F/_ w ( 2nd ` ( 1st ` x ) ) |
7 |
|
nfsbc1v |
|- F/ w [. ( 2nd ` x ) / w ]. ph |
8 |
6 7
|
nfsbcw |
|- F/ w [. ( 2nd ` ( 1st ` x ) ) / z ]. [. ( 2nd ` x ) / w ]. ph |
9 |
5 8
|
nfsbcw |
|- F/ w [. ( 1st ` ( 1st ` x ) ) / y ]. [. ( 2nd ` ( 1st ` x ) ) / z ]. [. ( 2nd ` x ) / w ]. ph |
10 |
|
nfv |
|- F/ x ph |
11 |
|
sbcoteq1a |
|- ( x = <. <. y , z >. , w >. -> ( [. ( 1st ` ( 1st ` x ) ) / y ]. [. ( 2nd ` ( 1st ` x ) ) / z ]. [. ( 2nd ` x ) / w ]. ph <-> ph ) ) |
12 |
1 4 9 10 11
|
ralxp3f |
|- ( A. x e. ( ( A X. B ) X. C ) [. ( 1st ` ( 1st ` x ) ) / y ]. [. ( 2nd ` ( 1st ` x ) ) / z ]. [. ( 2nd ` x ) / w ]. ph <-> A. y e. A A. z e. B A. w e. C ph ) |