Theorem rexpim ≪ | index | src | ≫ 

theorem rexpim {x y: nat} (a: wff y) (p: wff x) (q: wff x y):
  $ E. y (a /\ (P. x p -> q)) -> (P. x p -> E. y (a /\ q)) $;
StepHypRefExpression
1 expim 
E. y (P. x p -> a /\ q) -> (P. x p -> E. y (a /\ q))
2 bian1 
a -> (a /\ q <-> q)
3 2 pimeq2d 
a -> ((P. x p -> a /\ q) <-> (P. x p -> q))
4 3 bi2a 
a /\ (P. x p -> q) -> (P. x p -> a /\ q)
5 4 eximi 
E. y (a /\ (P. x p -> q)) -> E. y (P. x p -> a /\ q)
6 1, 5 syl 
E. y (a /\ (P. x p -> q)) -> (P. x p -> E. y (a /\ 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)