Step |
Hyp |
Ref |
Expression |
1 |
|
renegeulemv.b |
|- ( ph -> B e. RR ) |
2 |
|
renegeulemv.1 |
|- ( ph -> E. y e. RR ( B + y ) = A ) |
3 |
|
simprl |
|- ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) -> y e. RR ) |
4 |
|
simplrr |
|- ( ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) /\ x e. RR ) -> ( B + y ) = A ) |
5 |
4
|
eqcomd |
|- ( ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) /\ x e. RR ) -> A = ( B + y ) ) |
6 |
5
|
eqeq2d |
|- ( ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) /\ x e. RR ) -> ( ( B + x ) = A <-> ( B + x ) = ( B + y ) ) ) |
7 |
|
simpr |
|- ( ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) /\ x e. RR ) -> x e. RR ) |
8 |
|
simplrl |
|- ( ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) /\ x e. RR ) -> y e. RR ) |
9 |
1
|
ad2antrr |
|- ( ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) /\ x e. RR ) -> B e. RR ) |
10 |
|
readdcan |
|- ( ( x e. RR /\ y e. RR /\ B e. RR ) -> ( ( B + x ) = ( B + y ) <-> x = y ) ) |
11 |
7 8 9 10
|
syl3anc |
|- ( ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) /\ x e. RR ) -> ( ( B + x ) = ( B + y ) <-> x = y ) ) |
12 |
6 11
|
bitrd |
|- ( ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) /\ x e. RR ) -> ( ( B + x ) = A <-> x = y ) ) |
13 |
12
|
ralrimiva |
|- ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) -> A. x e. RR ( ( B + x ) = A <-> x = y ) ) |
14 |
|
reu6i |
|- ( ( y e. RR /\ A. x e. RR ( ( B + x ) = A <-> x = y ) ) -> E! x e. RR ( B + x ) = A ) |
15 |
3 13 14
|
syl2anc |
|- ( ( ph /\ ( y e. RR /\ ( B + y ) = A ) ) -> E! x e. RR ( B + x ) = A ) |
16 |
2 15
|
rexlimddv |
|- ( ph -> E! x e. RR ( B + x ) = A ) |