Theorem sb6f

Description: Equivalence for substitution when y is not free in ph . The implication "to the left" is sb2 and does not require the non-freeness hypothesis. Theorem sb6 replaces the non-freeness hypothesis with a disjoint variable condition and uses less axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb6f.1	$$⊢Ⅎy\phi$$
Assertion	sb6f	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sb6f.1	$$⊢Ⅎy\phi$$
2	1	nf5ri	$$⊢\phi \to \forall y\phi$$
3	2	sbimi	$$⊢\left[y/x\right]\phi \to \left[y/x\right]\forall y\phi$$
4		sb4a	$$⊢\left[y/x\right]\forall y\phi \to \forall x\left(x=y\to \phi \right)$$
5	3 4	syl	$$⊢\left[y/x\right]\phi \to \forall x\left(x=y\to \phi \right)$$
6		sb2	$$⊢\forall x\left(x=y\to \phi \right)\to \left[y/x\right]\phi$$
7	5 6	impbii	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$