Theorem ssab ≪ | index | src | ≫ 

theorem ssab {x: nat} (p q: wff x): $ A. x (p -> q) <-> {x | p} C_ {x | q} $;
StepHypRefExpression
1 nfv 
F/ y p -> q
2 nfab1 
FS/ x {x | p}
3 2 nfel2 
F/ x y e. {x | p}
4 nfab1 
FS/ x {x | q}
5 4 nfel2 
F/ x y e. {x | q}
6 3, 5 nfim 
F/ x y e. {x | p} -> y e. {x | q}
7 elab 
y e. {x | p} <-> [y / x] p
8 sbq 
x = y -> (p <-> [y / x] p)
9 7, 8 syl6bbr 
x = y -> (p <-> y e. {x | p})
10 elab 
y e. {x | q} <-> [y / x] q
11 sbq 
x = y -> (q <-> [y / x] q)
12 10, 11 syl6bbr 
x = y -> (q <-> y e. {x | q})
13 9, 12 imeqd 
x = y -> (p -> q <-> y e. {x | p} -> y e. {x | q})
14 1, 6, 13 cbvalh 
A. x (p -> q) <-> A. y (y e. {x | p} -> y e. {x | q})
15 14 conv subset 
A. x (p -> q) <-> {x | p} C_ {x | 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), axs_set (elab, ax_8)