Theorem sb6x

Description: Equivalence involving substitution for a variable not free. Usage of this theorem is discouraged because it depends on ax-13 . Usage of sb6 is preferred, which requires fewer axioms. (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb6x.1	$$⊢Ⅎx\phi$$
Assertion	sb6x	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sb6x.1	$$⊢Ⅎx\phi$$
2	1	sbf	$$⊢\left[y/x\right]\phi \leftrightarrow \phi$$
3		biidd	$$⊢x=y\to \left(\phi \leftrightarrow \phi \right)$$
4	1 3	equsal	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \phi$$
5	2 4	bitr4i	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$