Theorem sbequ12a

Description: An equality theorem for substitution. (Contributed by NM, 2-Jun-1993) (Proof shortened by Wolf Lammen, 23-Jun-2019)

	Ref	Expression
Assertion	sbequ12a	$$⊢x=y\to \left(\left[y/x\right]\phi \leftrightarrow \left[x/y\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbequ12r	$$⊢x=y\to \left(\left[x/y\right]\phi \leftrightarrow \phi \right)$$
2		sbequ12	$$⊢x=y\to \left(\phi \leftrightarrow \left[y/x\right]\phi \right)$$
3	1 2	bitr2d	$$⊢x=y\to \left(\left[y/x\right]\phi \leftrightarrow \left[x/y\right]\phi \right)$$