eq.sub

Theorem.

Arguments:

ca (coll), cb (coll),

Dummy variables:

x (sv),

Distinct variable conditions:

(x, ca), (x, cb),

Assertions:

((caequivcb)leftrightarrow((casubseteqcb)wedge(cbsubseteqca)))

Proof:

Hyp Ref Line Expr
df-sub1((casubseteqcb)leftrightarrowforallx((xinca)to(xincb)))
df-sub2((cbsubseteqca)leftrightarrowforallx((xincb)to(xinca)))
1, 2bi.bldan<>3(((casubseteqcb)wedge(cbsubseteqca))leftrightarrow(forallx((xinca)to(xincb))wedgeforallx((xincb)to(xinca))))
al.dfbi4(forallx((xinca)leftrightarrow(xincb))leftrightarrow(forallx((xinca)to(xincb))wedgeforallx((xincb)to(xinca))))
3, 4><bitr5(forallx((xinca)leftrightarrow(xincb))leftrightarrow((casubseteqcb)wedge(cbsubseteqca)))
df-equ6((caequivcb)leftrightarrowforallx((xinca)leftrightarrow(xincb)))
5, 6bitr7((caequivcb)leftrightarrow((casubseteqcb)wedge(cbsubseteqca)))