Theorem cbvexd | index | src |

theorem cbvexd {x y: nat} (G: wff) (p: wff x) (q: wff y):
  $ G /\ x = y -> (p <-> q) $ >
  $ G -> (E. x p <-> E. y q) $;
StepHypRefExpression
1 cbvexs
E. x p <-> E. y [y / x] p
2 1 a1i
G -> (E. x p <-> E. y [y / x] p)
3 sbet
A. x (x = y -> (p <-> q)) -> ([y / x] p <-> q)
4 hyp h
G /\ x = y -> (p <-> q)
5 4 ialda
G -> A. x (x = y -> (p <-> q))
6 3, 5 syl
G -> ([y / x] p <-> q)
7 6 exeqd
G -> (E. y [y / x] p <-> E. y q)
8 2, 7 bitrd
G -> (E. x p <-> E. y q)

Axiom use

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