Theorem wl-sbid2ft

Description: A more general version of sbid2vw . (Contributed by Wolf Lammen, 14-May-2019)

	Ref	Expression
Assertion	wl-sbid2ft	$$⊢Ⅎx\phi \to \left(\left[y/x\right]\left[x/y\right]\phi \leftrightarrow \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sb6	$$⊢\left[y/x\right]\left[x/y\right]\phi \leftrightarrow \forall x\left(x=y\to \left[x/y\right]\phi \right)$$
2		nfnf1	$$⊢ℲxℲx\phi$$
3		id	$$⊢Ⅎx\phi \to Ⅎx\phi$$
4		sbequ12r	$$⊢x=y\to \left(\left[x/y\right]\phi \leftrightarrow \phi \right)$$
5	4	a1i	$$⊢Ⅎx\phi \to \left(x=y\to \left(\left[x/y\right]\phi \leftrightarrow \phi \right)\right)$$
6	2 3 5	wl-equsaldv	$$⊢Ⅎx\phi \to \left(\forall x\left(x=y\to \left[x/y\right]\phi \right)\leftrightarrow \phi \right)$$
7	1 6	bitrid	$$⊢Ⅎx\phi \to \left(\left[y/x\right]\left[x/y\right]\phi \leftrightarrow \phi \right)$$