Theorem sb6a

Description: Equivalence for substitution. (Contributed by NM, 2-Jun-1993) (Proof shortened by Wolf Lammen, 23-Sep-2018)

	Ref	Expression
Assertion	sb6a	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \left[x/y\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbcov	$$⊢\left[y/x\right]\left[x/y\right]\phi \leftrightarrow \left[y/x\right]\phi$$
2		sb6	$$⊢\left[y/x\right]\left[x/y\right]\phi \leftrightarrow \forall x\left(x=y\to \left[x/y\right]\phi \right)$$
3	1 2	bitr3i	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \left[x/y\right]\phi \right)$$