Theorem expim ≪ | index | src | ≫ 

theorem expim {x y: nat} (p: wff x) (q: wff x y):
  $ E. y (P. x p -> q) -> (P. x p -> E. y q) $;
StepHypRefExpression
1 anl 
E. x p /\ A. x (p -> q) -> E. x p
2 1 conv pim 
(P. x p -> q) -> E. x p
3 2 eximi 
E. y (P. x p -> q) -> E. y E. x p
4 excom 
E. y E. x p -> E. x E. y p
5 id 
p -> p
6 5 eex 
E. y p -> p
7 6 eximi 
E. x E. y p -> E. x p
8 4, 7 rsyl 
E. y E. x p -> E. x p
9 3, 8 rsyl 
E. y (P. x p -> q) -> E. x p
10 exral 
E. y A. x (p -> q) -> A. x (p -> E. y q)
11 anr 
E. x p /\ A. x (p -> q) -> A. x (p -> q)
12 11 conv pim 
(P. x p -> q) -> A. x (p -> q)
13 12 eximi 
E. y (P. x p -> q) -> E. y A. x (p -> q)
14 10, 13 syl 
E. y (P. x p -> q) -> A. x (p -> E. y q)
15 9, 14 iand 
E. y (P. x p -> q) -> E. x p /\ A. x (p -> E. y q)
16 15 conv pim 
E. y (P. x p -> q) -> (P. x p -> E. y q)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12)